Chapter 10, Problem 4CR

(a)

To determine

The domain and the range of the function $y=3^x+2$

Answer to Problem 4CR

Solution:

The domain of the function $y=3^x+2$ is $\left(-\infty ,\infty \right)$ and range is $\left(2,\infty \right)$

Explanation of Solution

Given information:

The function $y=3^x+2$

Explanation:

The domain of $f$ is the set of possible $x$ values for which $f\left(x\right)$ is a real and defined.

The function $y=3^x+2$ is an exponential function.

The domain of the exponential function is the set of all real numbers $\left(-\infty ,\infty \right)$

Therefore, the domain of the given exponential function is $\left(-\infty ,\infty \right)$

The range of a function is the set of dependent variables for which the function is defined.

The range of exponential function is all real numbers greater than zero.

Then the range of $3^x$ is $\left(0,\infty \right)$

For all values of $x$, $3^x>0$

Then, $3^x+2>2$

Hence, the range of $y=3^x+2$ is $\left(2,\infty \right)$

(b)

To determine

The inverse of $y=3^x+2$ and the domain and the range of the inverse function.

Answer to Problem 4CR

Solution:

The inverse of $y=3^x+2$ is $y={\log }_3\left(x-2\right)$ and the domain is $\left(2,\infty \right)$ and the range is $\left(-\infty ,\infty \right)$

Explanation of Solution

Given information:

The function $y=3^x+2$

Explanation:

To find the inverse function $y=3^x+2$, first interchange the variables

$x=3^y+2$

Now, solve $x=3^y+2$ for $y$

$\Rightarrow x-2=3^y$

Applying logarithm on both sides

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

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

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

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

The inverse function of $y=3^x+2$ is $y={\log }_3\left(x-2\right)$

The domain of $f$ is the set of possible $x$ values, for which $f\left(x\right)$ is real and defined.

The function $y={\log }_3\left(x-2\right)$ is a logarithmic function.

The logarithm function is valid on the positive real axis.

The domain of the logarithmic function is all real numbers greater than zero, such as $\left(0,\infty \right)$

Then, $x-2>0$

$\Rightarrow x>2$

Therefore, the domain of the given function $y={\log }_3\left(x-2\right)$ is $\left(2,\infty \right)$

The range of a function is the set of dependent variables for which the function is defined.

The range of the logarithmic function is all real numbers, which is $\left(-\infty ,\infty \right)$

Hence, the range of $y={\log }_3\left(x-2\right)$ is $\left(-\infty ,\infty \right)$