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15

Expert-verifiedFound in: Page 176

Book edition
6th

Author(s)
Sullivan

Pages
1200 pages

ISBN
9780321795465

For the graph of the function f,

(a) Find the domain and the range of f.

(b) Find the intercepts.

(c) Is the graph of f symmetric with respect to the x-axis, the y-axis, or the origin?

(d) Find f (2).

(e) For what value(s) of x is $f\left(x\right)=3$?

(f) Solve $f\left(x\right)<0$.

(g) Graph $y=f\left(x\right)+2$.

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

(i) Graph $y=2f\left(x\right)$.

(j) Is f even, odd, or neither?

(k) Find the interval(s) on which f is increasing.

(a) The domain of f is ${D}_{f}=\left[-4,4\right]$ and the range is $R=\left[-1,3\right]$

(b) The intercepts are $\left(-1,0\right),\left(0,-1\right),\left(1,0\right)$

(c) The x coordinates of the corresponding pairs are symmetric with respect to the y-axis, so the f is symmetric with respect to the y-axis.

(d) $f\left(2\right)=1$

(e) The values of x are $\left(4,3\right)\to f\left(4\right)=3\phantom{\rule{0ex}{0ex}}\left(-4,3\right)\to f\left(-4\right)=3$

(f) localid="1650165486015" $x\in \left[-1,1\right]$

(g) The graph is

(h) The graph is

(i) The graph is

(j) The function f is even

(k) The function f is increasing $x\in \left[0,4\right]$

For the graph of the function *f*.

The domain is a set of the numbers x for which the function is defined. ${D}_{f}=\left[-4,4\right]$

The range is a set of all the values of the function, ${R}_{f}=\left[0,3\right]$

To find the intercept, check where the graph of the function f intersects the x-axis and the y--axis.

The intercepts are $\left(-1,0\right),\left(0,-1\right),\left(1,0\right)$

To find if the graph of f is symmetric with respect to the x-axis, the y-axis, or the origin check the different values of x, the function has the same value,

$\left(-4,3\right),\left(4,3\right)\phantom{\rule{0ex}{0ex}}\left(-2,1\right),\left(2,1\right)\phantom{\rule{0ex}{0ex}}\left(-1,0\right),\left(1,0\right)$

We can notice that the x- coordinate of the corresponding pairs is symmetric with respect to the y-axis, so the f is symmetric with respect to the y-axis.

To find the value of $f\left(2\right)$ find the value of the function if $x=2$

$f\left(2\right)=1$

As per the graph, we can notice that the solution are:

$\left(4,3\right)\to f\left(4\right)=3\phantom{\rule{0ex}{0ex}}\left(-4,3\right)\to f\left(-4\right)=3$

To find the values of x when the values of the function are less than zero, we have to find where is the graph of the function f under x-axis.

As per the graph $x\in \left[-1,1\right]$

If a constant k is added to the function, then there will be a vertical shift in the graph that is $k>0$.

If $k<0$ then the graph will move k unit downwards.

As per the question, we add 2 to the given function, therefore the graph will be

To graph $y=f\left(-x\right)$, we must reflect the graph of f along the y-axis.

The graph of f is symmetrical to the y-axis, therefore, the graph of $y=f\left(-x\right)$ should be the same with f

If a function is multiplied by a scalar factor k, then it will result to a vertical stretch to the graph of the function

if $k>1$, the graph is stretched vertically or narrow

If $0<k<1$ the graph is compressed or widens.

If, $k<0$, the graph is stretched and reflected along the x-axis

if the function is multiplied by positive 2, therefore the graph narrows or stretched vertically

The graph of $y=2f\left(x\right)$ is

The even functions are symmetric with respect to the y-axis and the odd functions are symmetric with respect to the origin. As per part (c) we have the graph of the f, is symmetric over the y-axis, so the function f is even.

If we increase the value of the argument x, the function values rise. So we have to find those z values.

As per the graph, the f is increasing:

$x\in \left[0,4\right]$

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