Answer :
To find the inverse of the function [tex]\( f(x) = 2x - 4 \)[/tex], follow these steps:
1. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
Let [tex]\( y = f(x) \)[/tex]. Hence, [tex]\( y = 2x - 4 \)[/tex].
2. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] and [tex]\( x \)[/tex] with [tex]\( y \)[/tex]:
This gives us the equation [tex]\( x = 2y - 4 \)[/tex].
3. Solve the equation for [tex]\( y \)[/tex]:
Add 4 to both sides to isolate the term with [tex]\( y \)[/tex]:
[tex]\[ x + 4 = 2y \][/tex]
4. Solve for [tex]\( y \)[/tex]:
Divide both sides by 2:
[tex]\[ y = \frac{x + 4}{2} \][/tex]
5. Simplify the expression:
[tex]\[ y = \frac{1}{2}x + 2 \][/tex]
Therefore, the inverse function [tex]\( g(x) \)[/tex] is given by:
[tex]\[ g(x) = \frac{1}{2}x + 2 \][/tex]
Comparing this with the given options:
- [tex]\( g(x) = \frac{1}{2}x - \frac{1}{4} \)[/tex]
- [tex]\( g(x) = \frac{1}{4}x - \frac{1}{2} \)[/tex]
- [tex]\( g(x) = 4x + 2 \)[/tex]
- [tex]\( g(x) = \frac{1}{2}x + 2 \)[/tex]
The correct inverse function is:
[tex]\[ g(x) = \frac{1}{2}x + 2 \][/tex]
So, the answer is:
[tex]\[ \boxed{4} \][/tex]
1. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
Let [tex]\( y = f(x) \)[/tex]. Hence, [tex]\( y = 2x - 4 \)[/tex].
2. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] and [tex]\( x \)[/tex] with [tex]\( y \)[/tex]:
This gives us the equation [tex]\( x = 2y - 4 \)[/tex].
3. Solve the equation for [tex]\( y \)[/tex]:
Add 4 to both sides to isolate the term with [tex]\( y \)[/tex]:
[tex]\[ x + 4 = 2y \][/tex]
4. Solve for [tex]\( y \)[/tex]:
Divide both sides by 2:
[tex]\[ y = \frac{x + 4}{2} \][/tex]
5. Simplify the expression:
[tex]\[ y = \frac{1}{2}x + 2 \][/tex]
Therefore, the inverse function [tex]\( g(x) \)[/tex] is given by:
[tex]\[ g(x) = \frac{1}{2}x + 2 \][/tex]
Comparing this with the given options:
- [tex]\( g(x) = \frac{1}{2}x - \frac{1}{4} \)[/tex]
- [tex]\( g(x) = \frac{1}{4}x - \frac{1}{2} \)[/tex]
- [tex]\( g(x) = 4x + 2 \)[/tex]
- [tex]\( g(x) = \frac{1}{2}x + 2 \)[/tex]
The correct inverse function is:
[tex]\[ g(x) = \frac{1}{2}x + 2 \][/tex]
So, the answer is:
[tex]\[ \boxed{4} \][/tex]
