Sure, let's find [tex]\( f(x) \)[/tex] and [tex]\( f(0) \)[/tex] given the functional equation [tex]\( f(x+2) = 2x + 1 \)[/tex].
### Step-by-Step Solution:
1. Assume the Form of [tex]\( f(x) \)[/tex]:
We start by assuming that [tex]\( f(x) \)[/tex] is a linear function, since the given functional equation [tex]\( f(x+2) = 2x + 1 \)[/tex] suggests linearity. Let:
[tex]\[
f(x) = ax + b
\][/tex]
2. Substitute [tex]\( x+2 \)[/tex] into the Assumed Form:
Substitute [tex]\( x + 2 \)[/tex] into the assumed form of [tex]\( f(x) \)[/tex]:
[tex]\[
f(x+2) = a(x+2) + b
\][/tex]
[tex]\[
f(x+2) = ax + 2a + b
\][/tex]
3. Set Up the Equation Using the Given Functional Equation:
According to the problem statement, [tex]\( f(x+2) = 2x + 1 \)[/tex]. So, equate the two expressions for [tex]\( f(x+2) \)[/tex]:
[tex]\[
ax + 2a + b = 2x + 1
\][/tex]
4. Solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
For the above equation to hold for all [tex]\( x \)[/tex], the coefficients of [tex]\( x \)[/tex] and the constant terms on both sides must be equal. Therefore, we get two separate equations:
[tex]\[
a = 2
\][/tex]
[tex]\[
2a + b = 1
\][/tex]
Substitute [tex]\( a = 2 \)[/tex] into the second equation:
[tex]\[
2(2) + b = 1
\][/tex]
[tex]\[
4 + b = 1
\][/tex]
[tex]\[
b = 1 - 4
\][/tex]
[tex]\[
b = -3
\][/tex]
5. Formulate [tex]\( f(x) \)[/tex] Using the Values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
Now that we have [tex]\( a = 2 \)[/tex] and [tex]\( b = -3 \)[/tex], we can write:
[tex]\[
f(x) = 2x - 3
\][/tex]
6. Find [tex]\( f(0) \)[/tex]:
Finally, substitute [tex]\( x = 0 \)[/tex] into [tex]\( f(x) \)[/tex] to find [tex]\( f(0) \)[/tex]:
[tex]\[
f(0) = 2(0) - 3
\][/tex]
[tex]\[
f(0) = -3
\][/tex]
### Summary:
- The function [tex]\( f(x) \)[/tex] is [tex]\( 2x - 3 \)[/tex].
- The value of [tex]\( f(0) \)[/tex] is [tex]\( -3 \)[/tex].
Thus, [tex]\( f(x) = 2x - 3 \)[/tex] and [tex]\( f(0) = -3 \)[/tex].
