Answer :
To find the inverse of the function \( f(x) = \frac{1}{4}x - 12 \), we will follow a systematic process step-by-step.
1. Express the function as \( y \) in terms of \( x \):
[tex]\[ y = \frac{1}{4}x - 12 \][/tex]
2. Swap \( x \) and \( y \):
[tex]\[ x = \frac{1}{4}y - 12 \][/tex]
3. Solve for \( y \) in terms of \( x \):
- First, isolate the term with \( y \):
[tex]\[ x + 12 = \frac{1}{4}y \][/tex]
- Next, multiply both sides of the equation by 4 to solve for \( y \):
[tex]\[ 4(x + 12) = y \][/tex]
- Simplify the equation:
[tex]\[ y = 4x + 48 \][/tex]
4. Write the inverse function:
The inverse function is:
[tex]\[ f^{-1}(x) = 4x + 48 \][/tex]
Next, we compare the inverse function with the given options:
- \( h(x) = 48x - 4 \)
- \( h(x) = 48x + 4 \)
- \( h(x) = 4x - 48 \)
- \( h(x) = 4x + 48 \)
The correct inverse of the function \( f(x) = \frac{1}{4}x - 12 \) is:
[tex]\[ h(x) = 4x + 48 \][/tex]
So the answer is:
[tex]\[ h(x) = 4 x + 48 \][/tex]
1. Express the function as \( y \) in terms of \( x \):
[tex]\[ y = \frac{1}{4}x - 12 \][/tex]
2. Swap \( x \) and \( y \):
[tex]\[ x = \frac{1}{4}y - 12 \][/tex]
3. Solve for \( y \) in terms of \( x \):
- First, isolate the term with \( y \):
[tex]\[ x + 12 = \frac{1}{4}y \][/tex]
- Next, multiply both sides of the equation by 4 to solve for \( y \):
[tex]\[ 4(x + 12) = y \][/tex]
- Simplify the equation:
[tex]\[ y = 4x + 48 \][/tex]
4. Write the inverse function:
The inverse function is:
[tex]\[ f^{-1}(x) = 4x + 48 \][/tex]
Next, we compare the inverse function with the given options:
- \( h(x) = 48x - 4 \)
- \( h(x) = 48x + 4 \)
- \( h(x) = 4x - 48 \)
- \( h(x) = 4x + 48 \)
The correct inverse of the function \( f(x) = \frac{1}{4}x - 12 \) is:
[tex]\[ h(x) = 4x + 48 \][/tex]
So the answer is:
[tex]\[ h(x) = 4 x + 48 \][/tex]
