Composition of Functions

Front

Define composition of functions.

Back

The composition of functions is a process where one function is applied to the result of another function. If \( f \) and \( g \) are two functions, the composition is denoted as \( (f \circ g)(x) = f(g(x)) \).

Front

What is the formula for finding \( f(g(x)) \) if \( f(x) = 2x \) and \( g(x) = 2x^2 - 1 \)?

Back

To find \( f(g(x)) \), substitute \( g(x) \) into \( f(x) \): \( f(g(x)) = f(2x^2 - 1) = 2(2x^2 - 1) = 4x^2 - 2 \).

Front

Calculate \( f(g(3)) \) for \( f(x) = 2x \) and \( g(x) = 2x^2 - 1 \).

Back

First, find \( g(3) = 2(3^2) - 1 = 17 \). Then, \( f(g(3)) = f(17) = 2(17) = 34 \).

Front

Calculate \( g(f(-1)) \) for \( f(x) = 2x \) and \( g(x) = 2x^2 - 1 \).

Back

First, find \( f(-1) = 2(-1) = -2 \). Then, \( g(f(-1)) = g(-2) = 2(-2)^2 - 1 = 7 \).

Front

What is the result of \( g(f(x)) \) if \( f(x) = 2x \) and \( g(x) = 2x^2 - 1 \)?

Back

To find \( g(f(x)) \), substitute \( f(x) \) into \( g(x) \): \( g(f(x)) = g(2x) = 2(2x)^2 - 1 = 8x^2 - 1 \).

Front

Define the notation \( g \circ f \).

Back

The notation \( g \circ f \) represents the composition of functions \( g \) and \( f \), meaning \( (g \circ f)(x) = g(f(x)) \).

Front

Calculate \( (g \circ f)(x) \) for \( f(x) = \frac{1}{x} \) and \( g(x) = 3x + 2 \).

Back

To find \( (g \circ f)(x) \), substitute \( f(x) \) into \( g(x) \): \( g(f(x)) = g\left(\frac{1}{x}\right) = 3\left(\frac{1}{x}\right) + 2 = \frac{3}{x} + 2 \).

Front

What is the output of \( f(g(x)) \) if \( f(x) = 2x \) and \( g(x) = 2x^2 - 1 \)?

Back

The output is \( f(g(x)) = f(2x^2 - 1) = 4x^2 - 2 \).

Front

What is the output of \( g(f(x)) \) if \( f(x) = 2x \) and \( g(x) = 2x^2 - 1 \)?

Back

The output is \( g(f(x)) = g(2x) = 8x^2 - 1 \).

Front

If \( f(x) = 2x \) and \( g(x) = 2x^2 - 1 \), what is \( f(g(0)) \)?

Back

First, find \( g(0) = 2(0)^2 - 1 = -1 \). Then, \( f(g(0)) = f(-1) = 2(-1) = -2 \).