To find the inverse of the given function [tex]\( f(x) = \frac{7x - 4}{2} \)[/tex], we will need to follow specific steps in algebra to solve for the inverse function.
1. Set [tex]\( y = f(x) \)[/tex]:
[tex]\[
y = \frac{7x - 4}{2}
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
Start by isolating [tex]\( x \)[/tex] in the given equation:
[tex]\[
2y = 7x - 4
\][/tex]
Next, solve for [tex]\( 7x \)[/tex]:
[tex]\[
7x = 2y + 4
\][/tex]
Finally, divide both sides by 7 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{2y + 4}{7}
\][/tex]
3. Express the inverse function [tex]\( f^{-1}(x) \)[/tex]:
By replacing [tex]\( y \)[/tex] with [tex]\( x \)[/tex] in the final equation, we get:
[tex]\[
f^{-1}(x) = \frac{2x + 4}{7}
\][/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] can be written as:
[tex]\[
f^{-1}(x) = \frac{2x + 4}{7}
\][/tex]
So, [tex]\( f^{-1}(x) \)[/tex] is [tex]\( \frac{2x + 4}{7} \)[/tex], where [tex]\( 2 \)[/tex], [tex]\( 4 \)[/tex], and [tex]\( 7 \)[/tex] are the coefficients of [tex]\( x \)[/tex], the constant term, and the denominator, respectively.
