Sure! Here's a step-by-step solution for finding the value of [tex]\( x \)[/tex] when [tex]\( f \circ g(x) = g^{-1}(x) \)[/tex], given the functions [tex]\( f(x) = 2x - 3 \)[/tex] and [tex]\( g(x) = 3x + 5 \)[/tex]:
1. Define the given functions:
- [tex]\( f(x) = 2x - 3 \)[/tex]
- [tex]\( g(x) = 3x + 5 \)[/tex]
2. Find the composition of [tex]\( f \)[/tex] and [tex]\( g \)[/tex], denoted as [tex]\( f \circ g(x) \)[/tex]:
- Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(g(x)) = f(3x + 5)
\][/tex]
- Now, calculate [tex]\( f(3x + 5) \)[/tex]:
[tex]\[
f(3x + 5) = 2(3x + 5) - 3 = 6x + 10 - 3 = 6x + 7
\][/tex]
- Therefore, [tex]\( f \circ g(x) = 6x + 7 \)[/tex].
3. Find the inverse of [tex]\( g(x) \)[/tex], denoted as [tex]\( g^{-1}(x) \)[/tex]:
- Start with the equation for [tex]\( g(x) \)[/tex]:
[tex]\[
y = 3x + 5
\][/tex]
- Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse:
[tex]\[
x = 3y + 5
\][/tex]
- Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{x - 5}{3}
\][/tex]
- Thus, [tex]\( g^{-1}(x) = \frac{x - 5}{3} \)[/tex].
4. Set the composition [tex]\( f \circ g(x) \)[/tex] equal to [tex]\( g^{-1}(x) \)[/tex]:
[tex]\[
6x + 7 = \frac{x - 5}{3}
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
- Eliminate the fraction by multiplying both sides of the equation by 3:
[tex]\[
3(6x + 7) = x - 5
\][/tex]
[tex]\[
18x + 21 = x - 5
\][/tex]
- Move all terms involving [tex]\( x \)[/tex] to one side and constant terms to the other side:
[tex]\[
18x - x = -5 - 21
\][/tex]
[tex]\[
17x = -26
\][/tex]
- Divide by 17 to isolate [tex]\( x \)[/tex]:
[tex]\[
x = -\frac{26}{17} = -\frac{13}{ \frac{17}{2}}
\][/tex]
- Therefore, the value of [tex]\( x \)[/tex] is:
[tex]\[
x = -\frac{19}{12}
\][/tex]
Thus, the value of [tex]\( x \)[/tex] is [tex]\( -\frac{19}{12} \)[/tex].
