Answer :
To find the inverse of the function [tex]\( f(x) = 4x \)[/tex], we need to follow a series of steps to determine which function [tex]\( h(x) \)[/tex], when applied to [tex]\( f(x) \)[/tex], gives us back the original input [tex]\( x \)[/tex].
### Steps to Find Inverse Function:
1. Rewrite the Function with [tex]\( y \)[/tex]:
Let's replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 4x \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
To find the inverse, we interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = 4y \][/tex]
3. Solve for [tex]\( y \)[/tex]:
We now solve for [tex]\( y \)[/tex] to get the expression for the inverse function:
[tex]\[ y = \frac{x}{4} \][/tex]
4. Rewrite the Inverse Function:
Replace [tex]\( y \)[/tex] with [tex]\( h(x) \)[/tex] to express the inverse function:
[tex]\[ h(x) = \frac{1}{4} x \][/tex]
### Conclusion:
The inverse function of [tex]\( f(x) = 4x \)[/tex] is [tex]\( h(x) = \frac{1}{4} x \)[/tex]. Among the given options, the correct answer is:
[tex]\[ h(x) = \frac{1}{4} x \][/tex]
### Steps to Find Inverse Function:
1. Rewrite the Function with [tex]\( y \)[/tex]:
Let's replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 4x \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
To find the inverse, we interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = 4y \][/tex]
3. Solve for [tex]\( y \)[/tex]:
We now solve for [tex]\( y \)[/tex] to get the expression for the inverse function:
[tex]\[ y = \frac{x}{4} \][/tex]
4. Rewrite the Inverse Function:
Replace [tex]\( y \)[/tex] with [tex]\( h(x) \)[/tex] to express the inverse function:
[tex]\[ h(x) = \frac{1}{4} x \][/tex]
### Conclusion:
The inverse function of [tex]\( f(x) = 4x \)[/tex] is [tex]\( h(x) = \frac{1}{4} x \)[/tex]. Among the given options, the correct answer is:
[tex]\[ h(x) = \frac{1}{4} x \][/tex]
