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Combination of Functions

Combination of Functions

The combination of functions is an important and fundamental part of algebraic mathematics. It is a skill that is applied across nearly all areas where mathematics is prevalent and is an indispensable concept in describing relationships between functions mathematically.

Definition of Combination of Functions

We can create a new function by adding, subtracting, multiplying, or dividing functions, just like how these operations form a new number.

The combination of functions is the act of combining multiple functions into a single function. This usually entails using basic mathematical operators such as addition or multiplication.

How to Combine Functions with Basic Operations

Functions can be combined using the same basic operators as any standard numerical set. These operators are:

As in other areas of math, the result of addition is the sum, the result of subtraction is the difference, the result of the multiplication is the product and the result of division is the quotient.

There are simple notation rules to follow when combining functions using these operators.

Addition

Subtraction

Multiplication

Division

Remember, we can only perform division when the denominator (the bottom of the fraction) is not zero. Why? Because dividing by zero is impossible!

There is another way that functions can be combined, known as the Composition of Functions. This entails forming whole a new function by using one function as the input to another. For instance, Confused? Well, not to worry, we have an explanation specially dedicated just to that topic!

How to Combine Functions by Addition

Consider the functionsand.

What happens when we combine them by addition?

Now we can substitute each of the functions.

And finally, collect like terms.

How to Combine Functions by Subtraction

Let's consider the same functions again, and.

What happens when we combine them by subtraction?

Now we substitute the functions as before.

And we can collect like terms.

How to Combine Functions by Multiplication

Let's consider the functions and

What happens if we multiply them together.

And substitute in the functions.

We then multiply out the brackets.

And again we can collect like terms.

How to Combine Functions by Division

Let's consider the functions and once again.

What happens when we combine them by division.

And substitute in the functions.

Then we simplify algebraically. In the case of these functions, we find a common factor and divide through.

How to Combine Functions in a Practical Situation

Let's take a look at a practical example of how functions get combined.

(1)

Below is a composite shape, a circle contained within a square. Each side of the square is a tangent to the circle. Find a function for the total shaded area of the shape in terms of the circle radius r.

Combination of functions area of a composite shape, StudySmarter A square with the area of a circle taken out of its centre, StudySmarter Originals

Solution:

Firstly, we must find the functions for the areas of the circle and the square.

We know the equation for the area of a circle is

We also can deduce that, if the length of each side of this square is the diameter of the circle, then the equation for the area of the square in terms of r is

We can define these as functions

Now, to find the function of the shaded area, we simply subtract the function for the area of the circle from the area of the square.

And then subbing in the functions we get

And once we simplify by expanding brackets and collecting like terms, we are left with a neat equation for the shaded area in terms of r in cm.

(2)

A store has a sale meaning that all clothes areoff. You wish to buy a pair of jeans that originally cost . The sales tax is . By constructing a composite function, find the final price of the jeans.

Solution:

To construct the composite function, we first must construct each constituent function. The first function, , will be the function for the price of the jeans after the discount, and the second function, , will be for the total price of the jeans.

Using as the input to we get the composite function

Where is the original cost of the jeans. Therefore the total cost of the jeans will be

How to Graph Combinations of Functions

Graphing combinations of functions can be done simply in two steps:

  1. Combine the relevant functions into a single function.

  2. Graph the newly created function.

What is really interesting, however, is seeing how combining two functions will alter the original graphs of those functions. For instance, if we combined the functions and , how do you think the graph of our combined function would compare to the graph ofLet's work through it and find out.

First, let's simply combine the functions and into a single function by addition.

And next as always we substitute in our functions and .

And as before, we simply gather like terms to find our final combined function.

Plotting the two original functions on a graph with the new combined function gives us the plot below. What can we notice about how relates to and Well, as we can see, has a greater gradient than either or In fact, the gradient of is equal to the gradients of and combined!

When we consider what we are really doing in combining functions, this makes perfect sense. The output of a combined function (the number on the vertical axis) for a given input (the number on the horizontal axis) is simply a combination of the outputs of each original function.

To test this, pick a value on the x-axis, and find the y-axis value for each of the functions. Does the y-axis value for equal the y-axis values for and added together?

composition of functions graph of combined functions, Studysmarter

Combination of two functions on a graph, John Hannah - StudySmarter Originals

Combination of Functions - Key takeaways

  • Functions can be combined using the basic operators, , in just the same way as numbers can.
  • There are many practical applications of combining functions, one of which is finding the function for the area of a composite shape.
  • The graph of a combined function is a combination of the graphs of its component functions.

Frequently Asked Questions about Combination of Functions

One way to combine functions is by addition, h(x) = g(x) + f(x).

Adding or subtracting linear functions from one another will always result in a linear function.

Combination of functions is a set of rules for merging multiple functions into one function in specific ways.

To find the combination of two functions, simply write a third function that combines the original two functions in the desired way, for instance, h(x) = f(x) - g(x), and substitute each of the original functions into the equation.  From there it is just a case of tidying up the new combined function to get its simplest form.

Final Combination of Functions Quiz

Question

What are the four basic operations for combining functions?

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Answer

Addition, Subtraction, Multiplication and Division.

Show question

Question

Complete the expression for the addition of functions:             

(f+g)(x) = ____

Show answer

Answer

(f+g)(x) = f(x) + g(x)

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Question

Complete the expression for the subtraction of functions: 


(f-g)(x) = ____



Show answer

Answer

(f-g)(x) = f(x) - g(x)

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Question

Complete the expression for the multiplication of functions:              

(fg)(x) = ____

Show answer

Answer

(fg)(x) = f(x) x g(x)

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Question

Complete the expression for the division of functions:


(f/g) = ____

Show answer

Answer

(f/g) = f(x) / g(x)

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Question

Given f(x) = 2x + 5, g(x) = 2x2 + 4 


find (f+g)(x)

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Answer

(f+g)(x) = 2x2 + 2x + 9

Show question

Question

Given f(x) = 2x + 5, g(x) = 2x2 + 4 


find (f-g)(x)

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Answer

(f-g)(x) = 2x - 2x2 + 1

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Question

Given f(x) = 2x + 5, g(x) = 2x2 + 4


find (fg)(x)

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Answer

(fg)(x)  = 4x3 + 10x2 + 8x + 20

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Question

Given f(x) = 2x + 5, g(y) = 3y2 + 3


find (f + g)(x,y)

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Answer

(f + g)(x,y) = 2x + 3y2 + 8 

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Question

What is composition of functions?

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Answer

Composition of functions is the method of using one function as the input to another, to form a third unique function.

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Question

Denote \[ h(x) = f(g(x)) \] using circle notation

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Answer

\[ h(x) = (f \circ g)(x) \]

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Question

In the composite function \( h(x) = (f \circ g)(x) \), which function is the inner function, and which function is the outer function?

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Answer

\( f(x) \) is the outer function and \( g(x) \) is the inner function.

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Question

Given f(x) = 2x + 5, g(y) = 3y2 + 3


find (f - g)(x,y)

Show answer

Answer

(f - g)(x,y) = 2x - 3y2 + 2

Show question

Question

Given f(x) = 2x + 5, g(y) = 3y2 + 3


find (fg)(x,y)

Show answer

Answer

(f - g)(x,y) = 6xy2 + 15y2 +6x + 15

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Question

What is a practical application for a combination of functions?

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Answer

Composite shapes.

Show question

Question

How can we form composite functions?

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Answer

By replacing each \( x \) (or alternative input variable) in the outer function, with the value of the inner function.

Show question

Question

Evaluate the composite function

\[ h(x) = f(f^{-1}(x))\]

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Answer

\[ h(x) = x \]

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Question

How is the domain of a composite function found?

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Answer

The domain of a composite function is the set of values in the domain of the inner function that are also within the domain of the outer function. 

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Question

Can a composite function be comprised of two functions with different variables?

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Answer

Yes

Show question

Question

Evaluate \[ h(x) = \sin^{-1}(\sin x) \]

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Answer

\[h(x) = x\]

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Question

Evaluate \[ h(x) = \sin (\sin^{-1} x) \]

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Answer

\[ h(x) = x \]

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Question

What is a composite function?

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Answer

A composite function is a function formed by one function used as the input to another.

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Question

Evaluate \[ h(x) = \sin^{-1}(\sin x) \]

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Answer

\[ h(x) = x \]

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Question

Decompose the following composite function into it's constituent functions.

\[ f(g(x)) = 2 \left( \frac{2}{x-1} \right) \] 

Show answer

Answer

\[ f(x) = 2x \]

\[ g(x) = \frac{2}{x-1} \]

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Question

Decompose the following composite function into it's constituent functions.

\[ (f \circ g)(x) = 1 - 2 \sqrt{2x + 3} \]

Show answer

Answer

\[ f(x) = 1 - 2 \sqrt{x} \]

\[ g(x) = 2x + 3 \]

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Question

Find the function \( h(x) = f(g(x)) \) given that \( f(x) = 2x \) and \( g(x) = x^2  \)

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Answer

\[ h(x) = 2x^2 \]

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Question

Find the function \( h(x) = f(g(x)) \) given that \( f(x) = \sqrt{x} \) and \( g(x) = 4x^2  \)

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Answer

\[ h(x) = 2x \]

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