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Question

Part 1
Find​ (a) ​(fcircle​g)(x) and​ (b) ​(gcircle​f)(x).
​f(x)equals7 x squared​; ​g(x)equalsStartFraction 1 Over x EndFraction
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Part 1
​(a) ​(fcircle​g)(x)equals
  
enter your response here ​(Simplify your​ answer.)

Answer 1

To find \((f \circ g)(x)\) and \((g \circ f)(x)\), we need to evaluate each function.

Given:

• \(f(x) = 7x^2\)
• \(g(x) = \frac{1}{x}\)

Part 1: Find \((f \circ g)(x)\)

The composition \((f \circ g)(x)\) means we will substitute \(g(x)\) into \(f(x)\):

1. Calculate \(g(x)\): \[ g(x) = \frac{1}{x} \]

2. Substitute \(g(x)\) into \(f(x)\): \[ (f \circ g)(x) = f(g(x)) = f\left(\frac{1}{x}\right) \]

3. Now plug \(\frac{1}{x}\) into \(f(x)\): \[ f\left(\frac{1}{x}\right) = 7\left(\frac{1}{x}\right)^2 \]

4. Simplify: \[ 7\left(\frac{1}{x}\right)^2 = 7 \cdot \frac{1}{x^2} = \frac{7}{x^2} \]

So, we have: \[ (f \circ g)(x) = \frac{7}{x^2} \]

Part 2: Find \((g \circ f)(x)\)

Now, we compute \((g \circ f)(x)\):

1. Substitute \(f(x)\) into \(g(x)\): \[ (g \circ f)(x) = g(f(x)) = g(7x^2) \]

2. Now plug \(7x^2\) into \(g(x)\): \[ g(7x^2) = \frac{1}{7x^2} \]

So, we have: \[ (g \circ f)(x) = \frac{1}{7x^2} \]

Final Answers

• \((f \circ g)(x) = \frac{7}{x^2}\)
• \((g \circ f)(x) = \frac{1}{7x^2}\)