Question

1. given the function $f(x) = \sqrt{-4x + 15}$, calculate the following values (round to two decimal places) and determine if the value is part of the domain of the function. (if an answer is not a real number, state that.)

a) $f(-7) =$
is the value of $-7$ in the domain of $f(x)$? yes or no?
b) $f(12) =$
is the value of $12$ in the domain of $f(x)$? yes or no?

Explanation:

Part (a)

Step 1: Substitute \( x = -7 \) into the function

We have \( f(x) = \sqrt{-4x + 15} \). Substituting \( x = -7 \), we get:
\( f(-7) = \sqrt{-4(-7) + 15} \)

Step 2: Simplify the expression inside the square root

First, calculate \( -4(-7) = 28 \). Then add 15: \( 28 + 15 = 43 \). So now we have \( f(-7) = \sqrt{43} \)

Step 3: Calculate the square root and round to two decimal places

\( \sqrt{43} \approx 6.56 \) (rounded to two decimal places)

Step 4: Determine the domain

For the square root function \( \sqrt{u} \), the expression inside the square root (\( u \)) must be non - negative (i.e., \( u\geq0 \)). We found that when \( x = -7 \), \( -4x + 15=43\geq0 \), so \( -7 \) is in the domain.

Step 1: Substitute \( x = 12 \) into the function

We have \( f(x)=\sqrt{-4x + 15} \). Substituting \( x = 12 \), we get:
\( f(12)=\sqrt{-4(12)+15} \)

Step 2: Simplify the expression inside the square root

First, calculate \( -4(12)=-48 \). Then add 15: \( -48 + 15=-33 \). So now we have \( f(12)=\sqrt{-33} \)

Step 3: Analyze the square root of a negative number

The square root of a negative number is not a real number. Also, since \( -4x + 15=-33<0 \) when \( x = 12 \), \( 12 \) is not in the domain of the function.

Answer:

\( f(-7)\approx6.56 \), and \( -7 \) is in the domain (Yes).

Part (b)