Questions: Copy and complete the table. g(x) f(x) (f ∘ g)(x) x/(x+2) x ? b. (x-1)/x (x)/(x+1) c. sqrt(x) x d. sqrt(x) ? x

Solution Steps

To solve the given problem, we need to compute the composition of functions \( (f \circ g)(x) \) for the provided functions \( g(x) \) and \( f(x) \). The composition \( (f \circ g)(x) \) means applying \( g(x) \) first and then applying \( f \) to the result of \( g(x) \).

1. Given \( g(x) = \frac{x}{x+2} \) and \( f(x) = |x| \), we need to find \( (f \circ g)(x) \).
2. Compute \( g(x) \) and then apply \( f \) to the result.

1. Given \( g(x) = \frac{x-1}{x} \) and \( f(x) = \frac{x}{x+1} \), we need to find \( (f \circ g)(x) \).
2. Compute \( g(x) \) and then apply \( f \) to the result.

1. Given \( g(x) = \sqrt{x} \) and \( f(x) = |x| \), we need to find \( (f \circ g)(x) \).
2. Compute \( g(x) \) and then apply \( f \) to the result.

1. Given \( g(x) = \sqrt{x} \) and \( (f \circ g)(x) = |x| \), we need to find \( f(x) \).
2. Determine \( f(x) \) such that \( f(\sqrt{x}) = |x| \).

Given: \[ g(x) = \frac{x}{x+2} \] \[ f(x) = |x| \]

To find \( (f \circ g)(x) \): \[ (f \circ g)(x) = f(g(x)) = f\left(\frac{x}{x+2}\right) = \left|\frac{x}{x+2}\right| \]

Given: \[ g(x) = \frac{x-1}{x} \] \[ f(x) = \frac{x}{x+1} \]

To find \( (f \circ g)(x) \): \[ (f \circ g)(x) = f(g(x)) = f\left(\frac{x-1}{x}\right) = \frac{\frac{x-1}{x}}{1 + \frac{x-1}{x}} = \frac{x-1}{x + (x-1)} = \frac{x-1}{2x-1} \]

Given: \[ g(x) = \sqrt{x} \] \[ f(x) = |x| \]

To find \( (f \circ g)(x) \): \[ (f \circ g)(x) = f(g(x)) = f(\sqrt{x}) = |\sqrt{x}| = \sqrt{x} \]

Given: \[ g(x) = \sqrt{x} \] \[ (f \circ g)(x) = |x| \]

We need to find \( f(x) \) such that: \[ f(\sqrt{x}) = |x| \]

There is no function \( f(x) \) that satisfies this equation for all \( x \). Therefore, the solution set is empty.

Final Answer