Sure, let's solve this problem step by step.
Given the functions:
[tex]\[ f(x) = 4x - 17 \][/tex]
[tex]\[ g(x) = \frac{2x + 8}{5} \][/tex]
We need to find [tex]\( x \)[/tex] such that:
[tex]\[ f(f(x)) = g^{-2}(x) \][/tex]
Step 1: Calculate [tex]\( f(f(x)) \)[/tex]
First, we'll compute [tex]\( f(f(x)) \)[/tex]:
[tex]\[ f(x) = 4x - 17 \][/tex]
Substitute [tex]\( f(x) \)[/tex] into itself:
[tex]\[ f(f(x)) = f(4x - 17) \][/tex]
[tex]\[ f(f(x)) = 4(4x - 17) - 17 \][/tex]
[tex]\[ f(f(x)) = 16x - 68 - 17 \][/tex]
[tex]\[ f(f(x)) = 16x - 85 \][/tex]
Step 2: Calculate [tex]\( g^2(x) \)[/tex] which is equivalent to [tex]\( g(g(x)) \)[/tex]
Now, we'll compute [tex]\( g(g(x)) \)[/tex]:
[tex]\[ g(x) = \frac{2x + 8}{5} \][/tex]
Substitute [tex]\( g(x) \)[/tex] into itself:
[tex]\[ g(g(x)) = g\left(\frac{2x + 8}{5}\right) \][/tex]
[tex]\[ g(g(x)) = \frac{2\left(\frac{2x + 8}{5}\right) + 8}{5} \][/tex]
Simplify inside the g function:
[tex]\[ g(g(x)) = \frac{\frac{4x + 16}{5} + 8}{5} \][/tex]
[tex]\[ g(g(x)) = \frac{\frac{4x + 16 + 40}{5}}{5} \][/tex]
[tex]\[ g(g(x)) = \frac{4x + 56}{25} \][/tex]
Step 3: Set [tex]\( f(f(x)) \)[/tex] equal to [tex]\( g(g(x)) \)[/tex]
Now, equate the two expressions:
[tex]\[ 16x - 85 = \frac{4x + 56}{25} \][/tex]
Step 4: Solve for [tex]\( x \)[/tex]
To solve this equation, we'll first eliminate the fraction by multiplying both sides by 25:
[tex]\[ 25(16x - 85) = 4x + 56 \][/tex]
[tex]\[ 400x - 2125 = 4x + 56 \][/tex]
Next, we'll isolate the terms involving [tex]\( x \)[/tex]:
[tex]\[ 400x - 4x = 56 + 2125 \][/tex]
[tex]\[ 396x = 2181 \][/tex]
Finally, solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{2181}{396} \][/tex]
Simplify the fraction:
[tex]\[ x = \frac{727}{132} \][/tex]
So, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\( f(f(x)) = g^{-2}(x) \)[/tex] is:
[tex]\[ x = \frac{727}{132} \][/tex]
