Questions: Suppose that (g(x)=7 x^2+5 x). Find (f(x)) so that the function can be described as (y=f(g(x))). (y=sqrt7 x^2+5 x) (f(x)=)

Suppose that (g(x)=7 x^2+5 x). Find (f(x)) so that the function can be described as (y=f(g(x))). (y=sqrt7 x^2+5 x) (f(x)=)
Transcript text: Suppose that $g(x)=7 x^{2}+5 x$. Find $f(x)$ so that the function can be described as $y=f(g(x))$. \[ \begin{array}{l} y=\sqrt{7 x^{2}+5 x} \\ f(x)=\square \end{array} \]
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Solution

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Solution Steps

To find \( f(x) \) such that \( y = f(g(x)) \), we need to express \( y \) in terms of \( g(x) \) and then identify the function \( f \). Given \( y = \sqrt{7x^2 + 5x} \) and \( g(x) = 7x^2 + 5x \), we can see that \( y = \sqrt{g(x)} \). Therefore, \( f(x) = \sqrt{x} \).

Step 1: Define the Functions

We are given the function \( g(x) = 7x^2 + 5x \) and need to find \( f(x) \) such that \( y = f(g(x)) \). The expression for \( y \) is given as \( y = \sqrt{7x^2 + 5x} \).

Step 2: Relate \( y \) to \( g(x) \)

Since \( g(x) = 7x^2 + 5x \), we can substitute \( g(x) \) into the expression for \( y \): \[ y = \sqrt{g(x)} \]

Step 3: Identify \( f(x) \)

From the relationship established in Step 2, we can express \( f(x) \) as: \[ f(x) = \sqrt{x} \]

Final Answer

Thus, the function \( f(x) \) that satisfies the condition is: \[ \boxed{f(x) = \sqrt{x}} \]

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