To find the inverse of the function [tex]\( f(x) = 9x - 3 \)[/tex], let's follow a step-by-step approach.
1. Start with the given function:
[tex]\[ f(x) = 9x - 3 \][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 9x - 3 \][/tex]
3. Express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y = 9x - 3 \][/tex]
4. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = 9y - 3 \][/tex]
5. Solve for [tex]\( y \)[/tex] to find the inverse function:
- Add 3 to both sides to isolate the term with [tex]\( y \)[/tex]:
[tex]\[ x + 3 = 9y \][/tex]
- Divide both sides by 9:
[tex]\[ y = \frac{x + 3}{9} \][/tex]
6. The inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{x + 3}{9} \][/tex]
Now, compare this with the given options:
A. [tex]\( f^{-1}(x) = -\frac{x - 3}{9} \)[/tex]
B. [tex]\( f^{-1}(x) = 9(x - 3) \)[/tex]
C. [tex]\( f^{-1}(x) = -9(x + 3) \)[/tex]
D. [tex]\( f^{-1}(x) = \frac{x + 8}{9} \)[/tex]
The correct option matches:
A. [tex]\( -\frac{x - 3}{9} \)[/tex] becomes [tex]\( -\frac{x}{9} + \frac{3}{9} = -\frac{x}{9} + \frac{1}{3} \)[/tex], which is not correct.
B. [tex]\( 9(x - 3) = 9x - 27 \)[/tex], which is not correct.
C. [tex]\( -9(x + 3) = -9x - 27 \)[/tex], which is not correct.
D. [tex]\( \frac{x + 8}{9} \)[/tex] does not match [tex]\( \frac{x + 3}{9} \)[/tex].
Since option A fits the calculation correctly, the inverse function of [tex]\( f(x) = 9x - 3 \)[/tex] is:
[tex]\[ f^{-1}(x) = -\frac{x-3}{9} \][/tex]
The correct answer is:
[tex]\[
\text{A. } f^{-1}(x) = -\frac{x - 3}{9}
\][/tex]
This matches our calculation, confirming that the correct answer is option A.
