Questions: Prove f(x) and g(x) are inverses both algebraically and graphically. 52. f(x) = 4/x - 2 g(x) = 4/(x+2)

Solution Steps

To prove that \( f(x) \) and \( g(x) \) are inverses both algebraically and graphically, we need to follow these steps:

1. Algebraic Proof:

   • Show that \( f(g(x)) = x \).
   • Show that \( g(f(x)) = x \).

2. Graphical Proof:

   • Plot the graphs of \( f(x) \) and \( g(x) \).
   • Plot the line \( y = x \) and check if the graphs of \( f(x) \) and \( g(x) \) are reflections over this line.

To prove that \( f(x) \) and \( g(x) \) are inverses algebraically, we need to show that \( f(g(x)) = x \) and \( g(f(x)) = x \).

Given: \[ f(x) = \frac{4}{x} - 2 \] \[ g(x) = \frac{4}{x+2} \]

Substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f\left( \frac{4}{x+2} \right) \] \[ f\left( \frac{4}{x+2} \right) = \frac{4}{\frac{4}{x+2}} - 2 \] \[ = \frac{4(x+2)}{4} - 2 \] \[ = x + 2 - 2 \] \[ = x \]

Thus, \( f(g(x)) = x \).

Substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g\left( \frac{4}{x} - 2 \right) \] \[ g\left( \frac{4}{x} - 2 \right) = \frac{4}{\left( \frac{4}{x} - 2 \right) + 2} \] \[ = \frac{4}{\frac{4}{x}} \] \[ = \frac{4x}{4} \] \[ = x \]

Thus, \( g(f(x)) = x \).

Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), we have shown algebraically that \( f(x) \) and \( g(x) \) are inverses.

To verify graphically, we need to plot both functions and check if they are reflections of each other across the line \( y = x \).

1. Plot \( f(x) = \frac{4}{x} - 2 \):

   • As \( x \to 0 \), \( f(x) \to \infty \) or \( f(x) \to -\infty \).
   • As \( x \to \infty \), \( f(x) \to -2 \).
   • As \( x \to -\infty \), \( f(x) \to -2 \).

2. Plot \( g(x) = \frac{4}{x+2} \):

   • As \( x \to -2 \), \( g(x) \to \infty \) or \( g(x) \to -\infty \).
   • As \( x \to \infty \), \( g(x) \to 0 \).
   • As \( x \to -\infty \), \( g(x) \to 0 \).

• If \( f(x) \) and \( g(x) \) are inverses, their graphs should be symmetric with respect to the line \( y = x \).

By plotting these functions, we can visually confirm that \( f(x) \) and \( g(x) \) are reflections of each other across the line \( y = x \).

Final Answer