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What is the inverse of the function [tex] f(x) = \frac{3-x}{7} [/tex]?

A. [tex] f^{-1}(x) = 3 - 7x [/tex]
B. [tex] f^{-1}(x) = \frac{7 + x}{3} [/tex]
C. [tex] f^{-1}(x) = 3 - \frac{x}{7} [/tex]
D. [tex] f^{-1}(x) = 7x - 3 [/tex]


Sagot :

To find the inverse of the function [tex]\( f(x) = \frac{3 - x}{7} \)[/tex], we need to follow these steps:

1. Rewrite the function using a different variable, typically [tex]\( y \)[/tex]:
[tex]\[ y = \frac{3 - x}{7} \][/tex]

2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. This involves rearranging the equation to isolate [tex]\( x \)[/tex].

Start by multiplying both sides by 7 to get rid of the fraction:
[tex]\[ 7y = 3 - x \][/tex]

Next, isolate [tex]\( x \)[/tex] by subtracting 3 from both sides:
[tex]\[ 7y - 3 = -x \][/tex]

Multiply both sides by -1 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = 3 - 7y \][/tex]

3. Interchange the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to write the inverse function. This means replacing [tex]\( y \)[/tex] with [tex]\( x \)[/tex] in the final expression from step 2:
[tex]\[ f^{-1}(x) = 3 - 7x \][/tex]

Therefore, the inverse of the function [tex]\( f(x) = \frac{3 - x}{7} \)[/tex] is:
[tex]\[ f^{-1}(x) = 3 - 7x \][/tex]

The correct answer is:
A. [tex]\( f^{-1}(x) = 3 - 7x \)[/tex]