Q.16 - Precalculus Enhanced with Graphing Utilities

Question

Use the graph of the function f shown to find:

(a) Find the domain and range of f.

(b) List the intercepts

(c) Find $f(-2)$

(d) Find the value(s) of x for which $f\left(x\right)=-3$

(e) Solve $f\left(x\right)>0.$

(f) Graph $y=f(x-3)$

(g) Graph $y=f\left(\frac{1}{2}x\right)$

(h) Graph  $y=-f\left(x\right)$

Step-by-Step Solution

Answer

(a) The domain of function is $\left\{x|-4\le x\le 3\right\}$ or the interval $[-4,3]$ and the range is $\left\{y|3\le x\le -3\right\}$ or the interval $[3,-3]$

(b)  Origin is the only intercept for the given function

(c)  $f(-2)=-1$

(d)  $f\left(x\right)=-3 when x=-4$

(e)  Inequality notations $f\left(x\right)>0 for 0\le x\le 3$

(f)  Graph for $y=f(x-3)$

(g) Graph for $y=f\left(\frac{1}{2}x\right)$

(h) Graph for $y=-f\left(x\right)$

Step 1.Given information

The given graph

Step 2.Find the domain and range of f.

The points on the graph of f  have  x-coordinates between -4  and 3 , inclusive. So, for each number x between -4 and 3 , there is a point $(x,f(x\left)\right)$ on the graph.

Therefore, the domain of f is $\left\{x|-4\le x\le 3\right\}$  or $[-4,3]$ the interval .

The points on the graph of f all have the y- coordinates between 3 and -3 , inclusive.
For each value of y between 3 and -3 , there exists at least one number x in the domain. 

 So, the range of f  is $\left\{y|3\le x\le -3\right\}$ or the interval $[3,-3]$ . 

Step 3.List the intercepts

Intercepts are the points where the graph touches the coordinate axes. The x-intercepts have the y-coordinates 0, and the y-intercepts have the x-coordinates  0.
From the graph we can see that the origin $(0,0)$  is the only intercept.

Step 4.Find f ( - 2 )

$f(-2)$ is the value of y on the graph when the value of x  is -2 .
The point $(-2,-1)$ on the graph corresponds to this situation. Thus, $f(-2)=-1$. 

Step 5.Find the value(s) of x for which f ( x ) = - 3

$f\left(x\right)=-3$means that the value of y  on the graph is -3 at a given x-value. Look for points on the graph for which the y-coordinate is -3 . 

The point $(-4,-3)$ on the graph corresponds to this situation.
Thus,$f\left(x\right)=-3$ , when $x=-4$

 

Step 6.Solve f ( x ) ≥ 0

 The function $f\left(x\right)>0$, indicates that the y-coordinate should have positive values for each value of the x-coordinate.
In the graph, x-values from -4 to 3 are shown. Determine for which of these values the y- coordinate is positive.
The points occur on $[0,3]$ .
So, using inequality notations, $f\left(x\right)>0$ for  $0\le x\le 3$

Step 7.Graph y = f ( x - 3 )

Since 3 is substracted from $f\left(x\right)$ ,the graph of $y=f(x-3)$ is obtained by shifting the graph of f horigontally to the right by 3 units

Step 8.Graph y = f 1 2 x

The graph of $f\left(\frac{1}{2}x\right)$is the horizontally compressed versionof the graph of $f\left(x\right)$ by a factor of $\frac{1}{2}$.The function $f\left(x\right)$ is graphed by multiplaying each x -coordinate of $f\left(x\right)$ by $\frac{1}{2}$

Step 9.Graph y = - f ( x )

The graph of the function $f\left(x\right)$ is the reflection of the graph of $f\left(x\right)$ about the y-axis.The function $f\left(x\right)$ is graphed by multiplaying the x -coordinates of $f\left(x\right)$  by -1.