Chapter 3, Problem 27REAE

Solution

To graph the function $y=\sqrt{x+3}+5$ and state the domain and range.

The graph of the function is,

  

Domain: $\left[-3,\infty \right)$

Range: $\left[5,\infty \right)$

Given information:

The function $y=\sqrt{x+3}+5$ .

Concept used:

• The graph of the function $y=f\left(x\right)$ and its inverse $f^{-1}\left(x\right)$ are symmetric to the line $y=x$ .
• For any point $\left(a,b\right)$ on the graph of $f\left(x\right)$ , the corresponding point on the graph of inverse function $f^{-1}\left(x\right)$ is $\left(b,a\right)$ .
• If a function is one-to-one, then it has an inverse.

Calculation:

The graph of the function $y=\sqrt{x+3}+5$ is shown below,

  

Since the domain of a function $y=f\left(x\right)$ is the set of x -values, the domain of the given function includes all real number that are greater than or equal to $-3$ , that is, the domain is $\left[-3,\infty \right)$ .

Since the range of a function $y=f\left(x\right)$ is the set of y -values, the range of the given function includes all real number that are greater than or equal to 5, that is, the domain is $\left[5,\infty \right)$ .