To solve the problem where [tex]\( f(x) = x^2 + 2 \)[/tex] and [tex]\( g(x) = \frac{1}{x - 2} \)[/tex], and we need to find [tex]\( x \)[/tex] such that [tex]\( g(f(x)) = 4 \)[/tex], follow these steps:
1. Compose the functions:
We need to find [tex]\( g(f(x)) \)[/tex]. Begin by substituting [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
f(x) = x^2 + 2
\][/tex]
Then,
[tex]\[
g(f(x)) = g(x^2 + 2) = \frac{1}{(x^2 + 2) - 2} = \frac{1}{x^2}
\][/tex]
2. Set up the equation:
We are given that [tex]\( g(f(x)) = 4 \)[/tex]. Substitute [tex]\( g(f(x)) \)[/tex]:
[tex]\[
\frac{1}{x^2} = 4
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
To find [tex]\( x \)[/tex], first solve the equation:
[tex]\[
\frac{1}{x^2} = 4
\][/tex]
Take the reciprocal of both sides to get:
[tex]\[
x^2 = \frac{1}{4}
\][/tex]
Now solve for [tex]\( x \)[/tex] by taking the square root of both sides:
[tex]\[
x = \pm \sqrt{\frac{1}{4}}
\][/tex]
Simplify the square root:
[tex]\[
x = \pm \frac{1}{2}
\][/tex]
4. Conclusion:
The solutions to the equation [tex]\( g(f(x)) = 4 \)[/tex] are:
[tex]\[
x = -\frac{1}{2} \quad \text{and} \quad x = \frac{1}{2}
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( g(f(x)) = 4 \)[/tex] are [tex]\( x = -\frac{1}{2} \)[/tex] and [tex]\( x = \frac{1}{2} \)[/tex].
