Question

2 2. For function h(x) - a. Find (-1) b. Find the domain of h(x), write your answer in set-builder notation. 2 c. For function (x) h'(x) and state the domain and range of h'(x). 3. a. What is the relationship between the graph of a function and the graph of its inverse? b. Graph the function (x) = V1 – 3 and its inverse on the same coordinate system. Determine at least 3 points on each graph. Label each function on your graph and state the domain and range of each. Note you do not need to find the inverse function, you are only asked to graph it. For f(x): Domain: Range: Page 2 ui 6 For f'(x): Domain: Range:

Answer 1

2.

Solution:

a.

h(x) = T +3

Replace
2
by  $\small -1$

2 ..h(-1)=3 -1+3 NIN = = 1

..h(-1)=1

b.

The domain is the set of input or argument values for which the function is real and defined.

Find undefined points:  

$\small x+3=0\Rightarrow x=-3$

The function domain is :

$\small \left \{ x\in \mathbb{R}\: |\: x<-3\: \: \mathrm{or}\: \: x>-3 \right \}$

$\small c.$

To find the inverse function,swap
2
and $\small y$ , and solve the resulting equation for $\small x.$

$\small y=\frac{2}{x+3}$ becomes $\small x=\frac{2}{y+3}$

Now, solve the equation  $\small x=\frac{2}{y+3}$ for $\small y$ .

$\small y=-3+\frac{2}{x}$

$\small \therefore h^{-1}\left ( x \right )=-3+\frac{2}{x}$

Singular point: $\small x=0$

Domain :  $\small \left ( -\infty ,0 \right )\cup \left ( 0,\infty \right )$

The function range is the combined domain of the inverse functions.

Inverse of the function :

$\small \frac{2}{x+3}$

Domain of the inverse function is:

$\small x<-3$ or $\small x>-3$

Combine the intervals:

$\small f\left ( x \right )<-3$ or   $\small f\left ( x \right )>-3$

i.e.  $\small \left ( -\infty ,-3 \right )\cup \left ( -3,\infty \right )$