Q 10 - Precalculus Enhanced with Graphing Utilities

Question

Use the given graph of the function f to answer parts (a) – (n).

(a) Find $f\left(0\right)$  and $f\left(6\right)$.

(b) Find $f\left(2\right)$ and $f\left(-2\right)$.

(c) Is $f\left(3\right)$ positive or negative?

(d) Is $f\left(-1\right)$ positive or negative?

(e) For what values of x is $f\left(x\right) = 0$?

(f) For what values of x is $f\left(x\right) <0$?

(g) What is the domain of f ?

(h) What is the range of f ?

(i) What are the x-intercepts?

(j) What is the y-intercept?

(k) How often does the line y = -1 intersect the graph?

(l) How often does the line x = 1 intersect the graph?

(m) For what value of x does $f\left(x\right) = 3$?

(n) For what value of x does $f\left(x\right) = -2$?

Step-by-Step Solution

Answer

(a) $$\begin{gathered} f\left(0\right)=0 \\ f\left(6\right)=0 \end{gathered}$$

(b) $f\left(2\right)=-2 and f\left(-2\right)=1$ 

(c) $f\left(3\right)$ is negative

(d) $f\left(-1\right)$ is positive

(e) $f\left(x\right)=0$ when $x=0,4,6$

(f)  $f\left(x\right)<0 for x\in \left(0,4\right)$

(g) $D:\left[-4,6\right]$

(h)  $R:\left[-2,3\right]$

(i) x-intercepts $\left(0,0\right), \left(4,0\right) and \left(6,0\right)$

(j) y-intercepts is $\left(0,0\right)$

(k) thrice

(l) once

(m) $x=5$

(n) $x=2$

Part (a) Step 1. Given information

Consider the given graph,

Part (a) of Step 1. Find f(0) and f(6)

Using the graph to determine the values we get

$$\begin{gathered} f\left(0\right)=0 \\ f\left(6\right)=0 \end{gathered}$$

Part (b) of Step 1. Find f(2) and f(-2)

When $x=2$, point $\left(2,-2\right)$ in on the graph, so $f\left(2\right)=-2$,

similarly when $x=-2$, point $\left(-2,1\right)$ so $f\left(-2\right)=1$

Part (c) of Step 1. f(3) is positive or negative

From the graph, we can see that value of $f\left(3\right)$ is below the x-axis so $f\left(3\right)$ is negative

Part (d) of Step 1. f(-1) positive or negative

From the graph, we can see that the value $f\left(-1\right)$ is above the x-axis, so $f\left(-1\right)$ is positive

Part (e) of Step 1. Value of x

From the graph, we can see that the points $\left(0,0\right), \left(4,0\right) and \left(6,0\right)$ are on the graph

$f\left(x\right)=0$ when $x=0,4,6$

Part (f) of Step 1. Value of x

To find the values of x for which $f\left(x\right)<0$,

we have to determine the x values for which the y-coordinates are negative $f\left(x\right)<0 for x\in \left(0,4\right)$

Part (g) of Step 1. Domain

The domain is the region where x exists on the function

$D:\left[-4.6\right]$

Part (h) of Step 1. Range

The range of a function is the set of all output values of a function.

From the given graph we can see that the range of function f is $R:\left[-2,3\right]$

Part (i) of Step 1. x-intercepts

x-intercepts are those points where $y=0$

From the graph, we have that x-intercepts are points $\left(0,0\right), \left(4,0\right) and \left(6,0\right)$

Part (j) of Step 1. y-intercepts

y-intercepts are points  at which $x=0$

From the graph, we can see that y-intercepts is $\left(0,0\right)$

Part (k) of Step 1.

From the graph, we can notice that the line $y=-1$ intersects the provided graph twice.

Part (l) of Step 1.

From the graph, we can notice that the line $x=1$ intersects the graph only once.

Part (m) of Step 1. value of x

From the graph we can see that when $x=5$ then $f\left(5\right)=3$

Therefore $x=5$

Part (n) of Step 1 . Value of x

From the graph we can see that $f\left(x\right)=-2$ when $x=2$