Questions: Let g(x) = x / (8-x) and h(x) = x^3 - 4x^2 + 4. Calculate the function g(h(t)). g(h(t)) =

Let g(x) = x / (8-x) and h(x) = x^3 - 4x^2 + 4. Calculate the function g(h(t)). g(h(t)) =
Transcript text: Let $g(x)=\frac{x}{8-x}$ and $h(x)=x^{3}-4 x^{2}+4$. Calculate the function $g(h(t))$. \[ g(h(t))= \]
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Solution

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Solution Steps

To find g(h(t)) g(h(t)) , we need to substitute h(t) h(t) into the function g(x) g(x) . This involves two steps:

  1. Calculate h(t) h(t) using the given function h(x)=x3−4x2+4 h(x) = x^3 - 4x^2 + 4 .
  2. Substitute the result of h(t) h(t) into g(x)=x8−x g(x) = \frac{x}{8-x} .
Step 1: Calculate h(t) h(t)

First, we need to calculate h(t) h(t) using the function h(x)=x3−4x2+4 h(x) = x^3 - 4x^2 + 4 . For t=2 t = 2 :

h(2)=23−4⋅22+4=8−16+4=−4 h(2) = 2^3 - 4 \cdot 2^2 + 4 = 8 - 16 + 4 = -4

Step 2: Substitute h(t) h(t) into g(x) g(x)

Next, we substitute h(t)=−4 h(t) = -4 into the function g(x)=x8−x g(x) = \frac{x}{8-x} :

g(−4)=−48−(−4)=−48+4=−412=−13≈−0.3333 g(-4) = \frac{-4}{8 - (-4)} = \frac{-4}{8 + 4} = \frac{-4}{12} = -\frac{1}{3} \approx -0.3333

Final Answer

−0.3333 \boxed{-0.3333}

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