Q 11 - Precalculus Enhanced with Graphing Utilities

Question

Consider the function, $f\left(x \right)= \frac{ x }{x + 4}$.

(a) Is the point, $\left(1,\frac{1}{4}\right)$ on the graph of $f$ ?

(b) If $x = -2$, 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 is on the graph of $f$ ?

Step-by-Step Solution

Answer

(a) The point is not on the graph of the function $f$

(b)  $f\left(-2\right)=-1$ and the graph of $f$ is $\left(-2,-1\right)$

(c) When $f\left(x\right)=2$ the value of the function is  $-4$ and the point on the graph of f is $\left(-4,2\right)$

Part (a) Step 1. Given information

Consider the function, $f\left(x\right)=\frac{x}{x+4}$

Part (a) Step 1. Check

Put the coordinates of the point

 $$\begin{gathered} P=\left(x,y\right) \\ P=\left(1,\frac{1}{4}\right) \end{gathered}$$

in the equation.

$\begin{array}{rcl}f\left(x\right)& =& \frac{x}{x+4}\\ \frac{1}{4}& =& \frac{1}{1+4}\\ \frac{1}{4}& \ne & \frac{1}{5}\end{array}$

So the point is not on the graph of the function f.

Part (b) Step 1. Find f x when x = - 2

Substitute $x=-2$ in the function

$$\begin{gathered} f\left(x\right)=\frac{x}{x+4} \\ f\left(-2\right)=\frac{-2}{-2+4} \\ f\left(-2\right)=-1 \end{gathered}$$

When , $x=-2$, the value of the function is $-1$. The graph of $f$ is $\left(-2,-1\right)$

Part (c) Step 1. Find x when f ( x ) = 2

Substitute $f\left(x\right)=2$ in the given equation and solve,

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

When $f\left(x\right)=2$, the value of the x is -4 the point on the graph of f is $\left(-4,2\right)$.