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Q. 26

Expert-verified
Found in: Page 171

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# In Problems 25–32, use the given functions f and g.$\left(a\right)f\left(x\right)=0\phantom{\rule{0ex}{0ex}}\left(b\right)g\left(x\right)=0\phantom{\rule{0ex}{0ex}}\left(c\right)f\left(x\right)=g\left(x\right)\phantom{\rule{0ex}{0ex}}\left(d\right)f\left(x\right)>0$$\left(e\right)g\left(x\right)\le 0\phantom{\rule{0ex}{0ex}}\left(f\right)f\left(x\right)>g\left(x\right)\phantom{\rule{0ex}{0ex}}\left(g\right)f\left(x\right)\ge 1$$f\left(x\right)=-{x}^{2}+3\phantom{\rule{0ex}{0ex}}g\left(x\right)=-3x+3$

The required solution sets are

(a) $x=-\sqrt{3},\sqrt{3}$

(b) $x=1$

(c) $x=0,3$

(d) $\left\{x:-\sqrt{3}

(e) $x\ge 1$

(f) $\left\{x:0

(g) $\left\{x:-\sqrt{2}\le x\le \sqrt{2}\right\}$

See the step by step solution

## Part (a) Step 1. Given information

• The given functions are $f\left(x\right)=-{x}^{2}+3\phantom{\rule{0ex}{0ex}}g\left(x\right)=-3x+3$.
• The equation is $f\left(x\right)=-{x}^{2}+3\phantom{\rule{0ex}{0ex}}g\left(x\right)=-3x+3$.

## Part (a) Step 2: Plot the graph and observe

• Plot the graph of the function.

• From the graph, it can be observed that $f\left(x\right)=0$ when $x=-\sqrt{3},\sqrt{3}$.

## Part (b) Step 1. Given information

• The given functions are $f\left(x\right)=-{x}^{2}+3\phantom{\rule{0ex}{0ex}}g\left(x\right)=-3x+3$
• The equation is $g\left(x\right)=0$.

## Part (b) Step 2. Plot the function and observe

• Plot the line in the graph obtained for the first function.

• From the graph, it can be observed that $g\left(x\right)=0$when $x=1$.

## Part (c) Step 1. Given information

• The given functions are $f\left(x\right)=-{x}^{2}+3\phantom{\rule{0ex}{0ex}}g\left(x\right)=-3x+3$
• The equation is $f\left(x\right)=g\left(x\right)$.

## Part (c) Step 2. Read the Graph

• For $f\left(x\right)=g\left(x\right)$, the curves of both the functions must intersect.
• From the graph, it can be observed that the functions intersect at $\left(0,3\right)and\left(3,-6\right)$.
• So, $f\left(x\right)=g\left(x\right)$at $x=0,3$.

## Part (d) Step 1. Given information

• The given functions are $f\left(x\right)=-{x}^{2}+3\phantom{\rule{0ex}{0ex}}g\left(x\right)=-3x+3$
• The inequality is $f\left(x\right)>0$.

## Part (d) Step 2.Find the region above the horizontal axis.

• The inequality holds when the curve of the function is above the horizontal axis.
• According to the graph obtained in step 2 of part (b), the curve is above the horizontal axis when $-\sqrt{3}.
• So, the solution set is $\left\{x:-\sqrt{3}.

## Part (e) Step 1. Given information

• The given functions are $f\left(x\right)=-{x}^{2}+3\phantom{\rule{0ex}{0ex}}g\left(x\right)=-3x+3$
• The inequality is $g\left(x\right)\le 0$.

## Part (e) Step 2. Find the region on or below the horizontal axis.

• $g\left(x\right)\le 0$ when the line of the function is on or below the horizontal axis.
• From the graph, the line is on or below the axis for $x\ge 1$.

## Part (f) Step 1. Given information

• The given functions are $f\left(x\right)=-{x}^{2}+3\phantom{\rule{0ex}{0ex}}g\left(x\right)=-3x+3$
• The inequality is $f\left(x\right)>g\left(x\right)$.

## Part (f) Step 2. Read the graph

• The inequality holds when the curve lies above the line on the graph.
• From the graph, it can be observed that the curve is above the line when $0.
• So, the solution set of the inequality is $\left\{x:0.

## Part (g) Step 1. Given information

• The given functions are $f\left(x\right)=-{x}^{2}+3\phantom{\rule{0ex}{0ex}}g\left(x\right)=-3x+3$
• The inequality is $f\left(x\right)\ge 1$.

## Part (g) Step 2. Read the graph

• The inequality holds when the curve lies above the value 1 on the vertical axis.
• From the graph, the curve is above 1 when $-\sqrt{2}\le x\le \sqrt{2}$.
• So, the solution set of the inequality is $\left\{x:-\sqrt{2}\le x\le \sqrt{2}\right\}$.