Chapter 5, Problem 12CR

(a)

To determine

The domain, range and horizontal asymptote of the graph of $g\left(x\right)$ by graphing the function $g\left(x\right)=3^x+2$ using transformation.

Answer to Problem 12CR

Solution:

Domain of $g\left(x\right)=3^x+2$ is $\left(-\infty ,\infty \right)$ .

Range of $g\left(x\right)=3^x+2$ is $\left(2,\infty \right)$ .

Horizontal asymptote is $y=2$ .

Graph is:

  

Explanation of Solution

Given information:

The function $g\left(x\right)=3^x+2$ .

The parent function for the transformation is $y=3^x$ .

The exponential function $g\left(x\right)=a^x,a>1$ has horizontal asymptote $x-$ axis $(y=0)$ , and the graph contains points $\left(0,1\right),\left(1,a\right)$ .

Since $a=3>1$ , the graph of function $y=3^x$ has horizontal asymptote $x-$ axis $(y=0)$ and contains points $\left(0,1\right),\left(1,3\right)$ .

Therefore, the graph of function $y=3^x$ is:

  

The transformation $y=g\left(x\right)+k$ shifts the graph of $y=g\left(x\right)$ up by $k$ units. Every point $\left(x,y\right)$ on graph of $y=g\left(x\right)$ shifts to $\left(x,y+k\right)$ .

To get the graph of $g\left(x\right)=3^x+2$ , shift the graph of $y=3^x$ up by $2$ units, and point $\left(0,1\right)$ will shift to $\left(0,3\right)$ . Also, horizontal asymptote $y=0$ will shift up by $2$ units. So, $y=2$ would be the horizontal asymptote for the function $g\left(x\right)=3^x+2$ .

Therefore, the graph of function $g\left(x\right)=3^x+2$ is:

  

Domain of the function is set of all $x$ values for which the function is defined.

From graph function, $g\left(x\right)=3^x+2$ is defined for all $x$ values. The domain of function $g\left(x\right)=3^x+2$ is the set of all real numbers.

In an interval notation, the domain of function $g\left(x\right)=3^x+2$ is $\left(-\infty ,\infty \right)$ .

Range of function is the set of all $y$ values function takes.

From graph, it is clear that the range of function is the set of all $y$ values greater than $2$ .

In an interval notation, the range of function $g\left(x\right)=3^x+2$ is $\left(2,\infty \right)$ .

The line $y=2$ is horizontal asymptote for the function $g\left(x\right)=3^x+2$ .

(b)

To determine

The inverse of $g\left(x\right)=3^x+2$ and its domain, range and vertical asymptote.

Answer to Problem 12CR

Solution:

The inverse function of $g\left(x\right)=3^x+2$ is $\underset{\_}{g^{-1}\left(x\right)={\log }_3\left(x-2\right)}$ .

Domain of function $g^{-1}\left(x\right)$ is $\underset{\_}{\left(2,\infty \right)}$ .

Range of function $g^{-1}\left(x\right)$ is $\underset{\_}{\left(-\infty ,\infty \right)}$ .

Vertical asymptote is line $\underset{\_}{x=2}$ .

Explanation of Solution

Given information:

The function $g\left(x\right)=3^x+2$ .

Let $y=g\left(x\right)$ .

In function $y=3^x+2$ , interchange the variables $x$ and $y$ .

  $\Rightarrow x=3^y+2$

Solve this equation for $y$ .

Subtract $2$ from both side of $x=3^y+2$ ,

  $\Rightarrow x-2=3^y$

This can also be written as $3^y=x-2$ .

Apply logarithm to the base $3$ on both sides of $3^y=x-2$ ,

  $\Rightarrow {\log }_3\left(3^y\right)={\log }_3\left(x-2\right)$

Apply power rule of logarithm $\log \left(a^n\right)=n\log \left(a\right)$ ,

  $\Rightarrow y{\log }_3\left(3\right)={\log }_3\left(x-2\right)$

Apply logarithmic rule ${\log }_a\left(a\right)=1$ ,

  $\Rightarrow y\cdot 1={\log }_3\left(x-2\right)$

  $\Rightarrow y={\log }_3\left(x-2\right)$

The inverse function is $g^{-1}\left(x\right)={\log }_3\left(x-2\right)$ .

To check $g^{-1}\left(g\left(x\right)\right)=x$ and $g\left(g^{-1}\left(x\right)\right)=x$ .

To check $g^{-1}\left(g\left(x\right)\right)=x$ , substitute $x=3^x+2$ in function $g^{-1}\left(x\right)$ ,

  $\Rightarrow g^{-1}\left(g\left(x\right)\right)={\log }_3\left(3^x+2-2\right)$

  $\Rightarrow g^{-1}\left(g\left(x\right)\right)={\log }_3\left(3^x\right)$

Using power rule of logarithm,

  $\Rightarrow g^{-1}\left(g\left(x\right)\right)=x{\log }_3\left(3\right)$

Since ${\log }_3\left(3\right)=1$ , $g^{-1}\left(g\left(x\right)\right)=x$ .

To check $g\left(g^{-1}\left(x\right)\right)=x$ , substitute $x={\log }_3\left(x-2\right)$ in function $g\left(x\right)$ ,

  $g\left(g^{-1}\left(x\right)\right)=3^{{\log }_3\left(x-2\right)}+2$

Apply the rule $a^{{\log }_{a}x}=x$ ,

  $\Rightarrow g\left(g^{-1}\left(x\right)\right)=x-2+2$

  $\Rightarrow g\left(g^{-1}\left(x\right)\right)=x$

So, $g^{-1}\left(g\left(x\right)\right)=x$ and $g\left(g^{-1}\left(x\right)\right)=x$ .

Therefore, the inverse function of $g\left(x\right)=3^x+2$ is $g^{-1}\left(x\right)={\log }_3\left(x-2\right)$ .

Range of function $g\left(x\right)$ is the domain of inverse function $g^{-1}\left(x\right)$ .

Since the range of function $g\left(x\right)$ is $\left(2,\infty \right)$ , the domain of function $g^{-1}\left(x\right)={\log }_3\left(x-2\right)$ is $\left(2,\infty \right)$ .

Domain of function $g\left(x\right)$ is the range of inverse function $g^{-1}\left(x\right)$ .

Since the domain of function $g\left(x\right)$ is $\left(-\infty ,\infty \right)$ , the range of function $g^{-1}\left(x\right)={\log }_3\left(x-2\right)$ is $\left(-\infty ,\infty \right)$ .

For logarithmic function $y={\log }_a\left(x\right),a>1$ , the vertical asymptote is line $x=0$ .

The graph of function $g^{-1}\left(x\right)={\log }_3\left(x-2\right)$ can also obtained by shifting graph of function $y={\log }_3\left(x\right)$ right by $2$ units.

Vertical asymptote of function $y={\log }_3\left(x\right)$ is $x=0$ , which will also shift right by $2$ units.

Therefore, the vertical asymptote of function $g^{-1}\left(x\right)={\log }_3\left(x-2\right)$ would be line $x=2$ .

(c)

To determine

To graph: The functions $g\left(x\right)=3^x+2$ and $g^{-1}\left(x\right)={\log }_3\left(x-2\right)$ on the same coordinate axes.

Explanation of Solution

Given information:

The functions $g\left(x\right)=3^x+2$ and $g^{-1}\left(x\right)={\log }_3\left(x-2\right)$ .

The graphs of functions $g\left(x\right)$ and its inverse function $g^{-1}\left(x\right)$ are symmetric with respect to line $y=x$ .

So, the graphs of functions $g\left(x\right)=3^x+2$ and $g^{-1}\left(x\right)={\log }_3\left(x-2\right)$ would be symmetric with respect to line $y=x$ .

The function $g\left(x\right)=3^x+2$ has horizontal asymptote line $y=2$ and contains the points $\left(0,3\right)$ and $\left(1,5\right)$ .

The function $g^{-1}\left(x\right)={\log }_3\left(x-2\right)$ has vertical asymptote line $x=2$ and contain points $\left(3,0\right)$ and $\left(5,1\right)$ because $\left(0,3\right)$ and $\left(1,5\right)$ are on the graph of $g\left(x\right)=3^x+2$ .

Therefore, the graph of $g\left(x\right)=3^x+2$ and $g^{-1}\left(x\right)={\log }_3\left(x-2\right)$ on the same coordinate axes is:

  

Interpretation:

The graph represents the functions $g\left(x\right)=3^x+2$ and $g^{-1}\left(x\right)={\log }_3\left(x-2\right)$ on the same co-ordinate axes. The $x-$ intercept of $g^{-1}\left(x\right)={\log }_3\left(x-2\right)$ is $\left(3,0\right)$ .