Q 112 - Precalculus Enhanced with Graphing Utilities

Question

Suppose that $F\left(x\right)={\log }_2(x+1)-3$.

(a) What is the domain of F?

(b) What is F(7)? What point is on the graph of F?

(c) If F(x)=-1, what is x? What point is on the graph of F?

(d) What is the zero of F?

Step-by-Step Solution

Answer

(a) The domain of F is $(-1,\infty )$.

(b) The value of F(7)=0.

(c) The value of x is 3 when F(x)=-1.

(d) The zero of F(x) is 7.

Step 1. Given information.

The following function $F\left(x\right)=\hspace{0.17em}{\log }_2\left(x+1\right)-3$ is given.

Step 2. The domain and the roots.

(a) Here, x+1=0

x>-1 .

So, the domain is (-1,0).

(b)

$$\begin{gathered} G\left(x\right)=\log { }_2( x+1)-3 \\ G\left(7\right)={\log }_2(7+1)-3 \\ =0 \\ \left(c\right){\log }_2( x+1)-3=3 \\ x+1=2^0 \\ x=-1 \\ \left(d\right) {\log }_2( x+1)-3=0 \\ {\log }_2( x+1)=3 \\ x+1=2^3 \\ x=7 \end{gathered}$$