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{:[f(z)=((1)/(3))^(z)],[h(z)=(z+4)/(z-2)]:} Evaluate. (h@f)(0)=

f(z)=(13)zh(z)=z+4z−2 \begin{array}{l} f(z)=\left(\frac{1}{3}\right)^{z} \\ h(z)=\frac{z+4}{z-2} \end{array} \newlineEvaluate.\newline(h∘f)(0)= (h \circ f)(0)=

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Q. f(z)=(13)zh(z)=z+4z−2 \begin{array}{l} f(z)=\left(\frac{1}{3}\right)^{z} \\ h(z)=\frac{z+4}{z-2} \end{array} \newlineEvaluate.\newline(h∘f)(0)= (h \circ f)(0)=
  1. Understanding composition of functions: Understand the composition of functions. The composition of two functions (h∘f)(x)(h \circ f)(x) means we first apply ff to xx, and then apply hh to the result of f(x)f(x).
  2. Evaluating f(0)f(0): Evaluate f(0)f(0).
    f(z)=(13)zf(z) = (\frac{1}{3})^z
    f(0)=(13)0f(0) = (\frac{1}{3})^0
    Since any non-zero number to the power of 00 is 11, we have:
    f(0)=1f(0) = 1
  3. Substituting f(0)f(0) into h(z)h(z): Substitute f(0)f(0) into h(z)h(z). Now we need to evaluate hh at the result of f(0)f(0), which is h(1)h(1). h(z)=z+4z−2h(z) = \frac{z + 4}{z - 2} h(1)=1+41−2h(1) = \frac{1 + 4}{1 - 2}
  4. Calculating h(1)h(1): Calculate h(1)h(1).
    h(1)=(1+4)(1−2)h(1) = \frac{(1 + 4)}{(1 - 2)}
    h(1)=5(−1)h(1) = \frac{5}{(-1)}
    h(1)=−5h(1) = -5
  5. Concluding the value of (h∘f)(0)(h \circ f)(0): Conclude the value of the composition (h∘f)(0)(h \circ f)(0).(h∘f)(0)(h \circ f)(0) is the value of hh at f(0)f(0), which we found to be h(1)=−5h(1) = -5. Therefore, (h∘f)(0)=−5(h \circ f)(0) = -5.

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