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Complete the inverse function for the function given below.

[tex]\[
\begin{array}{c}
f(x) = 2x - 4 \\
f^{-1}(x) = \frac{x + \square}{2}
\end{array}
\][/tex]

Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).

Sagot :

GinnyAnswer
To find the inverse of the given function [tex]\( f(x) = 2x - 4 \)[/tex], follow these steps:

1. Begin with the original function:
[tex]\[ y = 2x - 4 \][/tex]

2. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[ x = 2y - 4 \][/tex]

3. Solve this equation for [tex]\( y \)[/tex]:
[tex]\[ x + 4 = 2y \][/tex]

4. Divide both sides by 2 to isolate [tex]\( y \)[/tex]:
[tex]\[ y = \frac{x + 4}{2} \][/tex]

The inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{x + 4}{2} \][/tex]

To express this in the form [tex]\( f^{-1}(x) = 1x + b \)[/tex], let's rewrite it:
[tex]\[ f^{-1}(x) = \frac{1}{2}x + \frac{4}{2} \][/tex]
[tex]\[ f^{-1}(x) = \frac{1}{2}x + 2 \][/tex]

Therefore, the correct value to fill in the box is:
[tex]\[ f^{-1}(x) = 1x + \boxed{2} \][/tex]
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