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Chapter 3: Problem 71

Find a formula for \((f \circ g)(x)\) assuming that \(f\) and \(g\) are the indicated functions. \(f(x)=5^{3+2 x}\) and \(g(x)=\log _{5} x\)

Short Answer

Expert verified
The function composition \((f \circ g)(x)\) for the given functions \(f(x)=5^{3+2 x}\) and \(g(x)=\log _{5} x\) is: \((f \circ g)(x) = 5^3 \cdot x^2\).

Step by step solution

01

Function Composition

To find \((f \circ g)(x)\), we need to substitute \(g(x)\) into \(f(x)\). This means we will replace \(x\) in the expression for \(f(x)\) with the expression for \(g(x)\): \((f \circ g)(x) = f(g(x))\) For our given functions, this will be: \(f(g(x))=f(\log_5{x})\)
02

Substitute and Simplify

Now, we will substitute the expression for \(g(x)\), which is \(\log_5{x}\), into \(f(x)\): \(f(\log_5{x}) = 5^{3 + 2(\log_5{x})}\) Now, let's use the property of exponents to simplify this expression. Recall that for any positive integer \(a\) and \(b\), the property \(a^{b\log_a{x}}=x^b\) holds true: \(5^{3 + 2\log_5{x}} = 5^3 \cdot 5^{2\log_5{x}} = 5^3 \cdot x^2\)
03

Final Answer

Now, we have the formula for \((f\circ g)(x)\): \((f \circ g)(x) = 5^3 \cdot x^2\) In summary, given the functions \(f(x)=5^{3+2 x}\) and \(g(x)=\log _{5} x\), the function composition \((f \circ g)(x)\) is given by the formula: \((f \circ g)(x) = 5^3 \cdot x^2\)

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