Q. 96 - Precalculus Enhanced with Graphing Utilities

Question

Suppose that $F\left(x\right)={\left(\frac{1}{3}\right)}^x-3$.

(a) What is $F(-5)$? What point is on the graph of F ?

(b) If $F\left(x\right)=24$, what is x? What point is on the graph of F ?

(c) Find the zero of F.

Step-by-Step Solution

Answer

Part (a) $F(-5)=240$ and the point on the graph of F is $(-5,240)$.

Part (b) $x=-3$ and the point on the graph of F is $(-3,24)$.

Part (c) The zero of F is $x=-1$.

Part (a) Step 1. Substitute x = - 5 in F ( x ) = ( 1 3 ) x - 3 .

This gives

$$\begin{gathered} F(-5)={\left(\frac{1}{3}\right)}^{-5}-3 \\ =3^5-3 \\ =243-3 \\ =240 \end{gathered}$$

The point on the graph of the function F is $(-5,240)$.

Part (b) Step 2. Finding x using F ( x ) = 24 .

We have

$$\begin{gathered} F\left(x\right)=24 \\ {\left(\frac{1}{3}\right)}^x-3=24 \\ {\left(\frac{1}{3}\right)}^x=24+3 \\ {3}^{-x}=27 \\ {3}^{-x}=3^3 \end{gathered}$$

On comparing,

$x=-3$

The point on the graph of the function F is $(-3,24)$.

Part (c) Step 1. Substitute F ( x ) = 0

This gives

$$\begin{gathered} {\left(\frac{1}{3}\right)}^x-3=0 \\ {\left(\frac{1}{3}\right)}^x=3 \\ {3}^{-x}=3^1 \end{gathered}$$

On comparing, 

$x=-1$