Q. 25 - Precalculus Enhanced with Graphing Utilities

Question

$f\left(x\right)=\frac{x+2}{x-6}$

(a) Is the point $\left(3, 14\right)$ on the graph of $f$?

(b) If $x=4$, what is $f\left(x\right)$? What point is on the graph of $f$?

(c) If $f\left(x\right)=2$, what is $x$? What point(s) are on the graph of $f$?

(d) What is the domain of $f$?

(e) List the x-intercepts, if any, of the graph of $f$.

(f) List the y-intercept, if there is one, of the graph of $f$. 

Step-by-Step Solution

Answer

Part (a) No

Part (b) $f\left(4\right)=-3$; The point on the graph is $\left(4,-3\right)$.

Part (c) Value of $x$ is $14$; Point on the graph is $\left(14, 2\right)$.

Part (d) The domain is $\left\{x\overline{)x\ne 6}\right\}$.

Part (e) The x-intercept is $-2$.

Part (f) The y-intercept is $-\frac{1}{3}$.

Part (a) Step 1. Find the value of the function given point.

When $x=3$,

$\begin{array}{rcl}f\left(x\right)& =& \frac{x+2}{x-6}\\ f\left(3\right)& =& \frac{3+2}{3-6}\\ & =& \frac{5}{-3}\\ & =& -\frac{5}{3}\end{array}$

The point $\left(3,-\frac{5}{3}\right)$ is on the graph of $f$; the point $\left(3, 14\right)$ is not.

Part (b) Step 1. Find the value at 4.

If $x=4$,

$\begin{array}{rcl}f\left(x\right)& =& \frac{x+2}{x-6}\\ f\left(4\right)& =& \frac{4+2}{4-6}\\ & =& \frac{6}{-2}\\ & =& -3\end{array}$

The point $\left(4,-3\right)$ is on the graph.

Part (c) Step 1. Find the value of x .

If $f\left(x\right)=2$, then

$\begin{array}{rcl}f\left(x\right)& =& 2\\ \frac{x+2}{x-6}& =& 2\\ x+2& =& 2\left(x-6\right)\\ x+2& =& 2x-12\\ 2x-x& =& 2+12\\ x& =& 14\end{array}$

The point on the graph is $\left(14, 2\right)$.

Part (d) Step 1. Find the domain.

The domain of f is $\left\{x\overline{)x\ne 6}\right\}$, since $x=6$ results in division by 0.

Part (e) Step 1. Find x -intercepts.

The x-intercepts of the graph of $f$ are the real solutions of the equation $f\left(x\right)=0$ that are in the domain of $f$.

The real solution of the equation $f\left(x\right)=\frac{x+2}{x-6}$ is,

$\begin{array}{rcl}f\left(x\right)& =& 0\\ \frac{x+2}{x-6}& =& 0\\ x+2& =& 0\\ x& =& -2\end{array}$

 The x-intercept is $-2$.

Part (f) Step 1. Find y -intercepts.

The y-intercepts of the graph of $f$ are the real values of $f\left(0\right)$.

$\begin{array}{rcl}f\left(x\right)& =& \frac{x+2}{x-6}\\ f\left(0\right)& =& \frac{0+2}{0-6}\\ & =& \frac{2}{-6}\\ & =& -\frac{1}{3}\end{array}$

The y-intercept is $-\frac{1}{3}$.