Answer :
To find the inverse function of [tex]\( f(x) = 2x + 5 \)[/tex], we follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 2x + 5 \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to solve for the inverse function:
[tex]\[ x = 2y + 5 \][/tex]
3. Solve for [tex]\( y \)[/tex] to find the inverse function:
[tex]\[ x - 5 = 2y \implies y = \frac{x - 5}{2} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = \frac{x - 5}{2} \][/tex]
Now we need to find [tex]\( f^{-1}(8) \)[/tex]:
4. Substitute [tex]\( x = 8 \)[/tex] into the inverse function:
[tex]\[ f^{-1}(8) = \frac{8 - 5}{2} \][/tex]
5. 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. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 2x + 5 \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to solve for the inverse function:
[tex]\[ x = 2y + 5 \][/tex]
3. Solve for [tex]\( y \)[/tex] to find the inverse function:
[tex]\[ x - 5 = 2y \implies y = \frac{x - 5}{2} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = \frac{x - 5}{2} \][/tex]
Now we need to find [tex]\( f^{-1}(8) \)[/tex]:
4. Substitute [tex]\( x = 8 \)[/tex] into the inverse function:
[tex]\[ f^{-1}(8) = \frac{8 - 5}{2} \][/tex]
5. 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]
