To find the inverse of the function [tex]\( f(x) = 2x - 10 \)[/tex], follow these detailed steps:
### Step 1: Understand the function
The given function is [tex]\( f(x) = 2x - 10 \)[/tex]. We'll denote the inverse function as [tex]\( f^{-1}(x) \)[/tex].
### Step 2: Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]
Let's rewrite the function using [tex]\( y \)[/tex]:
[tex]\[ y = 2x - 10 \][/tex]
### Step 3: Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
To find the inverse, we interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = 2y - 10 \][/tex]
### Step 4: Solve for [tex]\( y \)[/tex]
Now, we solve this new equation for [tex]\( y \)[/tex]:
- Start by isolating the term involving [tex]\( y \)[/tex]:
[tex]\[ x + 10 = 2y \][/tex]
- Next, divide both sides by 2 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{x + 10}{2} \][/tex]
### Step 5: Write the inverse function
Since [tex]\( y = \frac{x + 10}{2} \)[/tex] represents the inverse function, we can denote it as:
[tex]\[ f^{-1}(x) = \frac{x + 10}{2} \][/tex]
### Step 6: Match with the given options
The final step is to compare [tex]\( f^{-1}(x) = \frac{x + 10}{2} \)[/tex] with the given multiple-choice options:
- [tex]\( h(x) = 2x - 5 \)[/tex]
- [tex]\( h(x) = 2x + 5 \)[/tex]
- [tex]\( h(x) = \frac{1}{2}x - 5 \)[/tex]
- [tex]\( h(x) = \frac{1}{2}x + 5 \)[/tex]
It's clear that the inverse function [tex]\( f^{-1}(x) = \frac{x + 10}{2} \)[/tex] matches the option:
[tex]\[ h(x) = \frac{1}{2}x + 5 \][/tex]
Thus, the inverse function is:
[tex]\[ h(x) = \frac{1}{2}x + 5 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{h(x)=\frac{1}{2} x+5} \][/tex]
