Question

1. if $f(x) = \frac{1}{4}x + 3$, what is the equation for $f^{-1}(x)$?\

\\(\boldsymbol{f^{-1}(x) = 4(x + 3)}\\)\
\\(\boldsymbol{f^{-1}(x) = 4(x - 3)}\\)\
\\(\boldsymbol{f^{-1}(x) = 4x - 3}\\)\
\\(\boldsymbol{f^{-1}(x) = 4x + 3}\\)

Explanation:

Step1: Intercambiar \(x\) e \(y\)

Dado \(y = \frac{1}{4}x + 3\), intercambiamos \(x\) e \(y\) para obtener \(x = \frac{1}{4}y + 3\).

Step2: Resolver para \(y\)

Restamos 3 de ambos lados: \(x - 3 = \frac{1}{4}y\).
Multiplicamos ambos lados por 4: \(y = 4(x - 3)\).
Así, la función inversa \(f^{-1}(x) = 4(x - 3)\).

Answer:

$f^{-1}(x) = 4(x - 3)$ (the second option, or more precisely, the option with $f^{-1}(x) = 4(x - 3)$)