Sure! Let's solve for [tex]\( f(x) \)[/tex] and then find [tex]\( f(5) \)[/tex].
Step 1: Identify the relationship given for [tex]\( f(x+1) \)[/tex]
We are given the function:
[tex]\[ f(x + 1) = 3x + 5 \][/tex]
Step 2: Express [tex]\( f(x) \)[/tex] in terms of a variable
To find [tex]\( f(x) \)[/tex], we need to shift the argument of the function back by 1 unit. Specifically, we want to replace [tex]\( x \)[/tex] with [tex]\( x-1 \)[/tex] in the given equation. This leads us to:
[tex]\[ f((x-1) + 1) = 3(x-1) + 5 \][/tex]
Step 3: Simplify the expression
Now simplify the right side:
[tex]\[ f(x) = 3(x - 1) + 5 \][/tex]
[tex]\[ f(x) = 3x - 3 + 5 \][/tex]
[tex]\[ f(x) = 3x + 2 \][/tex]
So, the function [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) = 3x + 2 \][/tex]
Step 4: Find [tex]\( f(5) \)[/tex]
Now that we have the function [tex]\( f(x) = 3x + 2 \)[/tex], we can find the value of [tex]\( f \)[/tex] at [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = 3(5) + 2 \][/tex]
[tex]\[ f(5) = 15 + 2 \][/tex]
[tex]\[ f(5) = 17 \][/tex]
Therefore, the two results are:
Function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = 3x + 2 \][/tex]
Value of [tex]\( f(5) \)[/tex]:
[tex]\[ f(5) = 17 \][/tex]
