# 27.1: Review of functions and graphs

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## Exercise $$\PageIndex{1}$$

Find all solutions of the equation: $$\big| 3x-9\big| =6$$

$$x = 1$$ or $$x = 5$$

## Exercise $$\PageIndex{2}$$

Find the equation of the line displayed below. $$y=-\dfrac{1}{2} x+1$$

## Exercise $$\PageIndex{3}$$

Find the solution of the equation: $$x^3-3x^2+2x-2=0$$ Approximate your answer to the nearest hundredth.

$$x \approx 2.52$$

## Exercise $$\PageIndex{4}$$

Let $$f(x)=x^2-2x+5$$. Simplify the difference quotient $$\dfrac{f(x+h)-f(x)}{h}$$ as much as possible.

$$2 x-2+2 h$$

## Exercise $$\PageIndex{5}$$

Consider the following graph of a function $$f$$. Find: domain of $$f$$, range of $$f$$, $$f(3)$$, $$f(5)$$, $$f(7)$$, $$f(9)$$.

domain $$D=[2,7]$$, range $$R=(1,4], f(3)=2, f(5)=2, f(7)=4, f(9)$$ is undefined

## Exercise $$\PageIndex{6}$$

Find the formula of the graph displayed below. $$f(x)=-x^{2}+2$$

## Exercise $$\PageIndex{7}$$

Let $$f(x)=5x+4$$ and $$g(x)=x^2+8x+7$$. Find the quotient $$\left(\dfrac f g\right)(x)$$ and state its domain.

$$\left(\dfrac{f}{g}\right)(x)=\dfrac{5 x+4}{x^{2}+8 x+7}=\dfrac{5 x+4}{(x+7)(x+1)}$$ has domain $$D=\mathbb{R}-\{-7,-1\}$$

## Exercise $$\PageIndex{8}$$

Let $$f(x)=x^2+\sqrt{x-3}$$ and $$g(x)=2x-3$$. Find the composition $$(f\circ g)(x)$$ and state its domain.

$$(f \circ g)(x)=4 x^{2}-12 x+9+\sqrt{2 x-6}$$ has domain $$D=[3, \infty)$$

## Exercise $$\PageIndex{9}$$

Consider the assignments for $$f$$ and $$g$$ given by the table below.

$\begin{array}{|c||c|c|c|c|c|} \hline x & 2 & 3 & 4 & 5 & 6 \\ \hline f(x) & 5 & 0 & 2 & 4 & 2 \\ \hline g(x) & 6 & 2 & 3 & 4 & 1 \\ \hline \end{array} \nonumber$

Is $$f$$ a function? Is $$g$$ a function? Write the composed assignment for $$(f\circ g)(x)$$ as a table.

$$f$$ and $$g$$ are both functions, and the composition is given by the table:

$$\begin{array}{|c||c|c|c|c|c|} \hline x & 2 & 3 & 4 & 5 & 6 \\ \hline \hline f(x) & 5 & 0 & 2 & 4 & 2 \\ \hline g(x) & 6 & 2 & 3 & 4 & 1 \\ \hline(f \circ g)(x) & 2 & 5 & 0 & 2 & \text { undef. } \\ \hline \end{array}$$

## Exercise $$\PageIndex{10}$$

Find the inverse of the function $$f(x)=\dfrac{1}{2x+5}$$.

$$f^{-1}(x)=\dfrac{1}{2 x}-\dfrac{5}{2}$$