Answer :
To find the inverse of the function \( f(x) = 4x + 7 \), we will follow these steps:
1. Express the function in terms of \( y \):
[tex]\[ y = 4x + 7 \][/tex]
2. Solve for \( x \) in terms of \( y \):
- First, isolate \( x \). Subtract 7 from both sides:
[tex]\[ y - 7 = 4x \][/tex]
- Next, divide both sides by 4:
[tex]\[ x = \frac{y - 7}{4} \][/tex]
3. Rewrite the equation in terms of the inverse function \( f^{-1}(x) \):
- Replace \( y \) with \( x \):
[tex]\[ f^{-1}(x) = \frac{x - 7}{4} \][/tex]
Therefore, the inverse function of \( f(x) = 4x + 7 \) is:
[tex]\[ f^{-1}(x) = \frac{x - 7}{4} \][/tex]
Now let's compare this with the given options:
A. \( f^{-1}(x) = 4x - 7 \)
B. \( f^{-1}(x) = \frac{x - 7}{4} \)
C. \( f^{-1}(x) = \frac{x - 4}{7} \)
D. \( f^{-1}(x) = 7 - 4x \)
The correct answer is:
[tex]\[ \boxed{\text{B}} \][/tex]
1. Express the function in terms of \( y \):
[tex]\[ y = 4x + 7 \][/tex]
2. Solve for \( x \) in terms of \( y \):
- First, isolate \( x \). Subtract 7 from both sides:
[tex]\[ y - 7 = 4x \][/tex]
- Next, divide both sides by 4:
[tex]\[ x = \frac{y - 7}{4} \][/tex]
3. Rewrite the equation in terms of the inverse function \( f^{-1}(x) \):
- Replace \( y \) with \( x \):
[tex]\[ f^{-1}(x) = \frac{x - 7}{4} \][/tex]
Therefore, the inverse function of \( f(x) = 4x + 7 \) is:
[tex]\[ f^{-1}(x) = \frac{x - 7}{4} \][/tex]
Now let's compare this with the given options:
A. \( f^{-1}(x) = 4x - 7 \)
B. \( f^{-1}(x) = \frac{x - 7}{4} \)
C. \( f^{-1}(x) = \frac{x - 4}{7} \)
D. \( f^{-1}(x) = 7 - 4x \)
The correct answer is:
[tex]\[ \boxed{\text{B}} \][/tex]
