To find the value of [tex]\( x \)[/tex] such that [tex]\( f(g(x)) = 0 \)[/tex], let's work through the problem step-by-step.
1. Understanding the functions:
- [tex]\( f(x) = 2x - 8 \)[/tex]
- [tex]\( g(x) = x^2 \)[/tex]
2. Determine [tex]\( f(g(x)) \)[/tex]:
- [tex]\( g(x) = x^2 \)[/tex].
- Substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]: [tex]\( f(g(x)) = f(x^2) \)[/tex].
3. Set up the equation [tex]\( f(x^2) = 0 \)[/tex]:
- Substitute [tex]\( x^2 \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(x^2) = 2(x^2) - 8 \][/tex].
- Set this equal to 0:
[tex]\[ 2(x^2) - 8 = 0 \][/tex].
4. Solve the equation for [tex]\( x \)[/tex]:
- Start by isolating [tex]\( x^2 \)[/tex]:
[tex]\[ 2x^2 - 8 = 0 \][/tex].
- Add 8 to both sides:
[tex]\[ 2x^2 = 8 \][/tex].
- Divide both sides by 2:
[tex]\[ x^2 = 4 \][/tex].
- Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \pm2 \][/tex].
So, the values of [tex]\( x \)[/tex] that satisfy [tex]\( f(g(x)) = 0 \)[/tex] are [tex]\( \boxed{2} \)[/tex] and [tex]\( \boxed{-2} \)[/tex].
