Questions: Express the function in the form f ∘ g ∘ h if g(x)=2^x and h(x)=x^2. H(x)=1-2^(x^2) f(x)=

Solution Steps

To express the function \( H(x) = 1 - 2^{x^2} \) in the form \( f \circ g \circ h \), we need to identify the functions \( f \), \( g \), and \( h \) such that \( H(x) = f(g(h(x))) \). We are given \( g(x) = 2^x \) and \( h(x) = x^2 \). We can see that \( h(x) \) is applied first, followed by \( g(x) \), and finally, \( f(x) \) should be such that it completes the expression to match \( H(x) \).

1. Start with \( h(x) = x^2 \).
2. Apply \( g(x) = 2^x \) to get \( g(h(x)) = 2^{x^2} \).
3. Determine \( f(x) \) such that \( f(g(h(x))) = 1 - 2^{x^2} \). This implies \( f(x) = 1 - x \).

Thus, \( f(x) = 1 - x \).

We are given the function \( H(x) = 1 - 2^{x^2} \) and need to express it in the form \( f \circ g \circ h \). We have:

• \( g(x) = 2^x \)
• \( h(x) = x^2 \)

1. Start with \( h(x) = x^2 \).
2. Apply \( g(x) \) to \( h(x) \): \( g(h(x)) = 2^{x^2} \).
3. Determine \( f(x) \) such that \( f(g(h(x))) = 1 - 2^{x^2} \). This implies \( f(x) = 1 - x \).

To verify, compute \( f(g(h(x))) \):

• \( h(3) = 3^2 = 9 \)
• \( g(9) = 2^9 = 512 \)
• \( f(512) = 1 - 512 = -511 \)

The output matches the expected result of \( H(3) = 1 - 2^{3^2} = -511 \).

Final Answer