Answer :
To find the inverse of the function [tex]\( f(x) = 4x \)[/tex], we need to find a function [tex]\( h(x) \)[/tex] such that when we apply [tex]\( h \)[/tex] to [tex]\( f(x) \)[/tex], we get back to [tex]\( x \)[/tex]. Here's the step-by-step process to find the inverse:
1. Start with the function:
[tex]\[ f(x) = 4x \][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 4x \][/tex]
3. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y = 4x \implies x = \frac{y}{4} \][/tex]
4. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to express the inverse function:
[tex]\[ h(x) = \frac{x}{4} = \frac{1}{4}x \][/tex]
Thus, the inverse function of [tex]\( f(x) = 4x \)[/tex] is:
[tex]\[ h(x) = \frac{1}{4} x \][/tex]
Among the given options, the correct representation of the inverse function is:
\[
h(x) = \frac{1}{4} x
\
1. Start with the function:
[tex]\[ f(x) = 4x \][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 4x \][/tex]
3. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y = 4x \implies x = \frac{y}{4} \][/tex]
4. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to express the inverse function:
[tex]\[ h(x) = \frac{x}{4} = \frac{1}{4}x \][/tex]
Thus, the inverse function of [tex]\( f(x) = 4x \)[/tex] is:
[tex]\[ h(x) = \frac{1}{4} x \][/tex]
Among the given options, the correct representation of the inverse function is:
\[
h(x) = \frac{1}{4} x
\
