Question

write an equation of the inverse for the one - to - one function as defined.
(a(x)=-2x + 9)
the equation of the inverse is (a^{-1}(x)=square).

Explanation:

Step1: Replace \( a(x) \) with \( y \)

We start with the function \( a(x)= -2x + 9 \). Replace \( a(x) \) with \( y \), so we get \( y=-2x + 9 \).

Step2: Swap \( x \) and \( y \)

To find the inverse, we swap the roles of \( x \) and \( y \). So the equation becomes \( x=-2y + 9 \).

Step3: Solve for \( y \)

First, subtract 9 from both sides: \( x - 9=-2y \). Then, divide both sides by -2: \( y=\frac{x - 9}{-2}=\frac{-x + 9}{2}=-\frac{1}{2}x+\frac{9}{2} \).

Step4: Replace \( y \) with \( a^{-1}(x) \)

Now, we replace \( y \) with \( a^{-1}(x) \) to get the inverse function. So \( a^{-1}(x)=-\frac{1}{2}x+\frac{9}{2} \) (or we can also write it as \( \frac{-x + 9}{2} \) by simplifying the numerator).

Answer:

\( a^{-1}(x)=\frac{-x + 9}{2} \) (or \( a^{-1}(x)=-\frac{1}{2}x+\frac{9}{2} \))