Answer :
To find the value of the inverse function [tex]\( f^{-1}(8) \)[/tex] given that [tex]\( f(x) = 2x + 5 \)[/tex], we follow these steps:
1. Define the function:
[tex]\[ f(x) = 2x + 5 \][/tex]
2. To find the inverse function [tex]\( f^{-1}(x) \)[/tex], we need to switch the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and solve for [tex]\( y \)[/tex]:
[tex]\[ y = 2x + 5 \][/tex]
3. First, solve for [tex]\( x \)[/tex]:
[tex]\[ y - 5 = 2x \][/tex]
4. Isolate [tex]\( x \)[/tex] by dividing both sides by 2:
[tex]\[ x = \frac{y - 5}{2} \][/tex]
5. Replace [tex]\( x \)[/tex] with [tex]\( f^{-1}(y) \)[/tex] and [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to get the inverse function:
[tex]\[ f^{-1}(x) = \frac{x - 5}{2} \][/tex]
6. Now we need to find [tex]\( f^{-1}(8) \)[/tex]:
[tex]\[ f^{-1}(8) = \frac{8 - 5}{2} \][/tex]
7. Simplify the expression:
[tex]\[ f^{-1}(8) = \frac{3}{2} \][/tex]
Therefore, the value of [tex]\( f^{-1}(8) \)[/tex] is [tex]\( \frac{3}{2} \)[/tex].
The correct answer is:
[tex]\[ \boxed{\frac{3}{2}} \][/tex]
1. Define the function:
[tex]\[ f(x) = 2x + 5 \][/tex]
2. To find the inverse function [tex]\( f^{-1}(x) \)[/tex], we need to switch the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and solve for [tex]\( y \)[/tex]:
[tex]\[ y = 2x + 5 \][/tex]
3. First, solve for [tex]\( x \)[/tex]:
[tex]\[ y - 5 = 2x \][/tex]
4. Isolate [tex]\( x \)[/tex] by dividing both sides by 2:
[tex]\[ x = \frac{y - 5}{2} \][/tex]
5. Replace [tex]\( x \)[/tex] with [tex]\( f^{-1}(y) \)[/tex] and [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to get the inverse function:
[tex]\[ f^{-1}(x) = \frac{x - 5}{2} \][/tex]
6. Now we need to find [tex]\( f^{-1}(8) \)[/tex]:
[tex]\[ f^{-1}(8) = \frac{8 - 5}{2} \][/tex]
7. Simplify the expression:
[tex]\[ f^{-1}(8) = \frac{3}{2} \][/tex]
Therefore, the value of [tex]\( f^{-1}(8) \)[/tex] is [tex]\( \frac{3}{2} \)[/tex].
The correct answer is:
[tex]\[ \boxed{\frac{3}{2}} \][/tex]
