Chapter 6.3, Problem 47HP

To determine

Write an equation for a square root function with a domain of $\left\{x\right|x\ge -4\}$ ,a range of $\left\{y\right|y\le 6\}$ ,and that passes through (5,3).

Answer to Problem 47HP

  $f(x)=6-\sqrt{x+4}$

Explanation of Solution

Given:

Domain = $\left\{x\right|x\ge -4\}$

Range = $\left\{y\right|y\le 6\}$

Passes through (5,3)

Calculation:

Since the domain is $[-4,\infty ]$ , so the vertex is shifted from origin to x = -4.

Hence , the function must contain $\sqrt{x+4}$ as the principal function $y=\sqrt{x}$ is shifted 4 units to the left.

Since the range is $(-\infty ,6]$ , we are dealing with the principal function $y=-\sqrt{x}$ .

So, we will now include the expression $-\sqrt{x+4}$ and since the principal function is shifted 6 units upward , the function is :

  $y=6-\sqrt{x+4}$

Check the function :

Since the function passes through the point (5,3) , it must satisfy the equation $y=6-\sqrt{x+4}$ :

Left hand side = 3

Right hand side = $$

  $6-\sqrt{5+4}=6-3=3$

Left hand side = Right hand side

So, the equation is satisfied.

So, the required equation for square root function is $f(x)=6-\sqrt{x+4}$