Question

(a) Find the domain and range of $y=3^x+2$

(b) Find the inverse of $y=3^x+2$and state its domain and range.

Short answer

Part(a) The domain and range of function is $(-\infty ,\infty )and(2,\infty )$respectively.

Part(b) The domain and range of the inverse of function is$(2,\infty )and(-\infty ,\infty )$

Step-by-step solution

Part(a) Step 1. Given Information

The given function is$y=3^x+2$

Part(a) Step 2. Explanation

The given function is defined for all the values of x, thus, the domain of the function is $(-\infty ,\infty )$

The exponential function cannot produce negative value for any value of x, the smallest value of $3^x$is zero. Thus, the minimum value of the function is $y=2$and can extend up to infinity.

Thus, the range of the given function islocalid="1648802965823" $(2,\infty )$

Part(b) Step 1. Explanation

First find the inverse of the given function.

$$\begin{gathered} y=3^x+2 \\ y-2=3^x \\ \ln (y-2)=\ln \left(3^x\right) \\ \ln (y-2)=x\ln 3 \\ x=\frac{\ln (y-2)}{\ln 3} \end{gathered}$$

Now, we will replace x with y and y with x to obtain a function in x that is the inverse of the function.

$y=\frac{\ln (x-2)}{\ln 3}$

The domain for the inverse of the function is the range of the given function and vice-versa.

Thus, the domain and range of the function islocalid="1648802416836" $(2,\infty )and(-\infty ,\infty )$respectively.