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Algebraic Functions

Algebraic Functions

You can have a mathematical problem involving both known and unknown values. For example, if you know that the age of your uncle John is twice your age plus four years, and you know that your age is 15, then you can use an algebraic function to work out the age of your uncle.

In this article, we will define what algebraic functions are, the different types of algebraic functions, how to identify algebraic and non-algebraic functions, touch on the differential calculus of algebraic functions, and work through some examples.

  • What are algebraic functions?
  • Types of algebraic functions
  • Algebraic functions vs. non-algebraic functions
  • Differential calculus with algebraic functions
  • Graphing algebraic functions
  • Algebraic functions example problems

What are Algebraic Functions?

As we learned from our Functions article, there are many classes of functions. One of those classes is algebraic functions.

An Algebraic Function is a function that involves only the algebraic operations: addition, subtraction, multiplication, division, powers, and roots.

We created this class of functions when we allowed for quotients and fractional powers in polynomial functions. If it weren't for these allowances, we would simply have polynomial functions! These additions to the polynomial functions give rise to the types of algebraic functions:

  • Polynomial functions
  • Rational functions
  • Power/Root functions
    • This includes functions with fractional powers because they can be written as roots. For example:

Examples of Algebraic Functions

Based on our definition of an algebraic function, let's list some examples of algebraic functions.

Again, note that algebraic functions include only the operations: , integer exponents and rational exponents.

Types of Algebraic Functions

The image below shows the types of algebraic functions.

Algebraic Functions types and examples StudySmarterThe types of algebraic functions – StudySmarter Originals

Polynomial Functions

Algebraic functions can take on the form of Polynomial Functions:

Where,

  • , , … , are all real number constants

  • is a positive integer

Some examples of polynomial algebraic functions are:

  • A linear function, like:

  • A quadratic function, like:

  • A cubic function, like:

  • A biquadratic/quartic function, like:

  • A quintic function, like:

Rational Functions

Algebraic functions can take on the form of Rational Functions:

Where,

  • and are polynomials.

  • .

Some examples of rational algebraic functions are:

Power Functions

Algebraic functions can take on the form of Power Functions:

Where,

  • and are any real numbers.

Some examples of power algebraic functions are:

  • The reciprocal function: (note that this is also a rational function).

Be careful here! While they look similar, an exponential function like , is not a power function! Power functions must have the variable as the base, while exponential functions have the variable as the exponent.

Root Functions

A specific type of power function is the Root Function:

What makes this a special kind of power function?

  • A root function is a power function with fractional exponents, where,

    • is any real number.

    • and are positive integers greater than 1.

    • does not equal.

Some examples of root algebraic functions are:

Algebraic Functions vs. Non-Algebraic Functions

So, if an algebraic function can include only the operations , exponents (and roots), then what about all those other functions we've learned about? Those are all non-algebraic functions!

Some examples of non-algebraic functions are:

  • Trigonometric functions, like:

  • Hyperbolic functions, like :

  • Exponential functions, like :

  • Logarithmic functions, like :

  • Absolute value functions, like :

Differential Calculus with Algebraic Functions

When it comes to graphing algebraic functions, sometimes we need to take Derivatives and Higher-Order Derivatives to find the critical points and inflection points of the graphs of these functions.

Derivatives of Algebraic Functions – Critical Points

If we want to find the critical points on the graph of an algebraic function, we must know how to take the derivative of the function. This can be found in our article on Derivatives.

What is a critical point?

Let be an interior point in the domain of. We say that is a critical point of the function if:

  1. exists and if either:
    1. or
    2. doesn't exist

f(c) MUST EXIST for x=c to be a critical point!

But what is a critical point, exactly?

Wherever we have a critical point of a function, there is either a horizontal tangent, a vertical tangent, a sharp turn, or a change in concavity at that point.

  • If the critical point is , we have a horizontal tangent at that point. This means that either:
    • The function has a maximum or minimum point at \( c \).
      • If we have \( f(x)=x^{2} \), then \( f'(0)=0 \) and \( x=0 \) is a critical point. At this critical point, we have a horizontal tangent and an absolute minimum at \( x=0 \).Algebraic Functions graph of a function with a critical point and a local minimum StudySmarterIf we have \( f(x)=x^{2} \), then \( f'(0)=0 \) and \( x=0 \) is a critical point. At this critical point, we have a horizontal tangent and an absolute minimum at \( x=0 \). — StudySmarter Originals
    • Or the function has an inflection point at \( c \).
      • If we have \( f(x)=x^{3} \), then \( f'(0)=0 \) and \( x=0 \) is a critical point. At this critical point, we have a horizontal tangent and a change in concavity, but no minimum or maximum.Algebraic Functions graph with horizontal tangent at critical point StudySmarterIf we have \( f(x)=x^{3} \), then \( f'(0)=0 \) and \( x=0 \) is a critical point. At this critical point, we have a horizontal tangent and a change in concavity, but no minimum or maximum. — StudySmarter Originals
  • If the critical point is , we have either:
    • A vertical tangent at point \( c \).
      • If we have \( f(x)=x^{\frac{1}{3}} \), then \( f'(0)=DNE \) and \( x=0 \) is a critical point. At this critical point, we have a vertical tangent and a change in concavity.
      • Algebraic Functions graph with vertical tangent at critical point StudySmarterIf we have \( f(x)=x^{\frac{1}{3}} \), then \( f'(0)=DNE \) and \( x=0 \) is a critical point. At this critical point, we have a vertical tangent and a change in concavity. — StudySmarter Originals
    • Or a sharp turn at point \( c \).
      • If we have \( f(x)=\sqrt[3]{x^{2}} \), then \( f'(0)=DNE \) and \( x=0 \) is a critical point. At this critical point, we have a sharp turn and a change in concavity.
      • Algebraic Functions graph with sharp turn at critical point StudySmarterIf we have \( f(x)=\sqrt[3]{x^{2}} \), then \( f'(0)=DNE \) and \( x=0 \) is a critical point. At this critical point, we have a sharp turn and a change in concavity. — StudySmarter Originals

To find a critical point of a function, we follow these steps:

  1. Find the Derivative of the function.

  2. Set the derivative equal to 0.

    1. Solve for x, if possible.

  3. Find all values of x (if any) where the derivative is undefined.

  4. All the values of x (if they are within the domain of the original function) found in steps 2 and 3 are the x-coordinates of the critical points.

    • Find the y-values of the critical points by plugging the x-values into the original function and solving for the y-coordinate.

Higher-Order Derivatives of Algebraic Functions – Inflection Points

If we want to find the inflection points on the graph of an algebraic function, we must know how to take the second derivative of the function. This can be found in our article on Higher-Order Derivatives.

What is an inflection point?

If is continuous at and changes concavity at , then the point is an inflection point of .

But what is an inflection point, exactly?

Wherever we have an inflection point on a function, that is where it changes from either:

  • Convex to Concave, or
  • Concave to Convex

And what do Concave and Convex mean? 1

  • Convex (also called Concave Upward or Convex Downward) is when the slope of a function is increasing.

  • Concave (also called Concave Downward or Convex Upward) is when the slope of a function is decreasing.

Note that the AP Calculus Exams (and likely your AP Calculus teacher) use the terms Concave Up and Concave Down.

Algebraic Functions the difference between convex and concave StudySmarterThe difference between concave up and concave down – StudySmarter Originals

So, how do we find where a function changes concavity?

  • We find the second derivative of the function.

    • If the second derivative of a function is positive, then it is Convex.

    • If the second derivative of a function is negative, then it is Concave.

Graphing Algebraic Functions

The graphs of all these types of algebraic functions vary widely from each other. However, the general procedure to graph an algebraic function is as follows:

  1. Calculate the x-intercepts by setting and solving for x.

  2. Calculate the y-intercepts by setting and solving for y.

  3. Find and plot any asymptotes.

  4. Find and plot any critical points.

  5. Find and plot any inflection points.

  6. Calculate some extra points along the curve until you get a good feel for how the curve looks.

  7. Plot all these points and draw the curve that connects them.

Graph the function:

Solution:

  1. Calculate the x-intercepts by setting and solving for x.

    • can be replaced with.

    • → set and cross-multiply to solve for.

    • → There are no x-intercepts.

  2. Calculate the y-intercepts by setting and solving for y.

    • can be replaced with.

    • → set and solve for.

    • → you can't divide by 0! There are no y-intercepts.

  3. Find and plot any asymptotes.

    • Since the expression as, there is a horizontal asymptote at the line (the x-axis).

    • Since the expression as, there is a vertical asymptote at the line (the y-axis).

  4. Find and plot any critical points.

    • Find the derivative of the function (check out our article on Derivatives for more info):

    • Set the derivative equal to 0, and solve for x.

      • → while the point where is a possible candidate for a critical point because , there are no critical points because we proved earlier that does not exist.

  5. Find and plot any inflection points.

    • Find the second derivative of the function (check out our article on Higher-Order Derivatives for more info):

    • Set the second derivative equal to 0, and solve for x.

      • → the second derivative is never 0, and it is undefined when. There are no inflection points because the original function is not defined at 0.

  6. Calculate some extra points along the curve until you get a good feel for how the curve looks.

    • x1/x
      -4-1/4
      -3-1/3
      -2-1/2
      -1-1
      0undefined
      11
      21/2
      31/3
      41/4
  7. Plot all these points and draw the curve that connects them.

    • Algebraic Functions graph reciprocal function StudySmarterThe graph of the reciprocal function 1/x – StudySmarter Originals

Finding Domain and Range of Algebraic Functions

Finding the domain and range of algebraic functions depends on the type of algebraic function we are considering.

For polynomial functions:

  • The domain for all polynomial functions is all real numbers:

  • The range for polynomial functions depends on both the order of the polynomial and the y-values of the graph.

    • If the order of the polynomial is odd, then the range is always .

    • If the order of the polynomial is even, then the range depends on the minimum and/or maximum y-value(s).

Find the domain and range of the function:

Solution:

Since this is a polynomial function:

  • The domain is all real numbers:.

To find the range, we recognize that this is the equation of a parabola. We can rewrite the equation in vertex form as:

Where,

  • is the leading coefficient
  • is the vertex of the parabola

Since the leading coefficient is positive, we know the parabola opens upward. This means the vertex is the lowest point of the parabola.

Therefore, the range is .

We can graph the function to double-check our work:

Algebraic Functions graph polynomial StudySmarterThe graph of a polynomial function – StudySmarter Originals

For rational functions:

  • Finding the domain of rational functions requires that we follow the rule that the denominator cannot equal 0.

  • Finding the range of rational functions requires that we:

    • First, solve the function for x

    • Then, apply the rule that the denominator cannot equal 0.

Note that if the polynomials within the rational function have orders higher than 2, this process quickly becomes difficult. To find the range of rational functions whose orders are 3 or higher, we would need more differential analysis.

Find the domain and range of the function:

Solution:

Since this is a rational function, we must follow the rule that the denominator cannot equal 0. Therefore, the domain is the set of all real numbers except where the denominator equals 0.

  • Domain:

To find the range, we need to find the values of y for which there exists a real number of x such that:

  1. Multiply both sides of the equation by.
  2. Move all terms with an x to the right of the equals sign. Move all terms without an x to the left of the equals sign.
  3. Factor out the x on the right side of the equation.
  4. Solve for x.
  5. If, the equation has no solution because that would set the denominator equal to 0. Therefore, the range is:
    • Range:

We can graph the function to double-check our work:

Algebraic Functions graph rational function StudySmarterThe graph of a rational function – StudySmarter Originals

For power/root functions:

  • Both domain and range of power and root functions depend on the exponent. We need to analyze each function on a case-by-case basis, as the domain and range of these functions are pretty “sensitive” to changes in exponents.

Find the domain and range of the function:

Solution:

To find the domain:

  1. Since this is a square root function, we need to keep what is inside the radical greater than or equal to 0.
    • So, .
  2. Factor what is under the radical.
    • → this inequality holds only if both terms are positive or both terms are negative.
      • For both terms positive:
        • Therefore, must be part of the domain.
      • For both terms negative:
        • There are no values of x that can satisfy both of these inequalities.
  3. Domain:

To find the range:

  1. We use the domain to find the range:
    • If the domain is , then .
    • Therefore, .
  2. Range:

We can graph the function to double-check our work:

Algebraic Functions graph root function StudySmarterThe graph of a root function – StudySmarter Originals

Algebraic Functions Example Problems

Which of the following are algebraic functions?

Solutions:

  1. This is NOT an algebraic function.
  2. This is an algebraic function.
  3. This is NOT an algebraic function.
  4. This is NOT an algebraic function.
  5. This is an algebraic function.
  6. This is an algebraic function.
  7. This is an algebraic function.
  8. This is an algebraic function.
  9. This is NOT an algebraic function.
  10. This is NOT an algebraic function.
  11. This is an algebraic function.
  12. This is an algebraic function.

Find the domain of each of the following:

Solutions:

  1. We can't divide by 0, so the domain is the set of values of x such that .
    • Domain:
  2. Again, we can't divide by 0, so we need to determine the values of x where the denominator is 0.
    1. for all real numbers of x.
      • Domain:
  3. Since the square root of a negative number is not a real number, we must determine the values of x that satisfy .
      • Domain:
  4. Here, the cube root is defined for all real numbers.
    • Domain:

Algebraic Functions – Key takeaways

  • Algebraic functions include only the algebraic operations:
    • Addition: +
    • Subtraction: -
    • Multiplication:
    • Division:
    • Powers (aka exponents):, where is any real number
    • Roots:, where is any real number and is a positive integer greater than 1.
  • The types of algebraic functions are:
    • Polynomial functions
    • Rational functions
    • Power/Root functions
  • Any function that contains trigonometric functions, hyperbolic functions, logarithms, variables in the power (or exponent), or an absolute value is not an algebraic function.
  • Related reading:

References

  1. https://math.stackexchange.com/questions/2364116/how-to-remember-which-function-is-concave-and-which-one-is-convex/2364123#2364123

Frequently Asked Questions about Algebraic Functions

An algebraic function is a function that involves only the algebraic operations: addition, subtraction, multiplication, division, rational powers, and roots.

Two examples of algebraic functions are rational functions and root functions.

The types of algebraic functions are:

  • Polynomial Functions
  • Rational Functions
  • Power/Root Functions

Solving algebraic functions usually involves finding their domains and/or ranges and graphing them.

Learning algebraic functions are important for several reasons:

  • Helps you develop problem-solving skills
  • Learn how to structure and organize problems
  • Learn how to approach problems that have no solution
  • Learn to think abstractly
  • Learn to visualize situations

Final Algebraic Functions Quiz

Question

What are the types of algebraic functions?

Show answer

Answer

Polynomial functions

Show question

Question

What is a non-algebraic function?

Show answer

Answer

Any function that involves something other than the algebraic operations of addition, subtraction, multiplication, division, powers, and roots is a non-algebraic function.

Show question

Question

What is the general procedure to graph an algebraic function?

Show answer

Answer

  1. Calculate the x-intercepts by setting y = 0 and solving for x.

  2. Calculate the y-intercepts by setting x = 0 and solving for y.

  3. Find and plot any asymptotes.

  4. Find and plot any critical points.

  5. Find and plot any inflection points.

  6. Calculate some extra points along the curve until you get a good feel for how the curve looks.

  7. Plot all these points and draw the curve that connects them.

Show question

Question

What rule do we need to follow to find the domain of a rational function?

Show answer

Answer

Finding the domain of rational functions requires that we follow the rule that the denominator cannot equal 0.

Show question

Question

How do we find the critical points of an algebraic function?

Show answer

Answer

To find a critical point of a function, we follow these steps:

  1. Find the derivative of the function.

  2. Set the derivative equal to 0.

    1. Solve for x, if possible.

  3. Find all values of x (if any) where the derivative is undefined.

  4. All the values of x (if they are within the domain of the original function) found in steps 2 and 3 are the x-coordinates of the critical points.

    • Find the y-values of the critical points by plugging the x-values into the original function and solving for the y-coordinate.

Show question

Question

What do Concave and Convex mean?

Show answer

Answer

  • Convex (also called Concave Upward or Convex Downward) is when the slope of a function is increasing.

  • Concave (also called Concave Downward or Convex Upward) is when the slope of a function is decreasing.

Show question

Question

How do we find where a function changes concavity?

Show answer

Answer

  • We find the second derivative of the function.

    • If the second derivative of a function is positive, then it is Convex.

    • If the second derivative of a function is negative, then it is Concave.

Show question

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