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 \( f(x) \) and \( g(x) \) 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 \( f(x) = 2x + 3 \) and \( g(x) = -3x - 4 \)?

Back

To find \( f(g(x)) \), substitute \( g(x) \) into \( f(x) \): \( f(g(x)) = f(-3x - 4) = 2(-3x - 4) + 3 = -6x - 8 + 3 = -6x - 5 \).

Front

How do you find \( f(g(x)) \) for \( f(x) = \frac{1}{x} \) and \( g(x) = 3x + 2 \)?

Back

Substitute \( g(x) \) into \( f(x) \): \( f(g(x)) = f(3x + 2) = \frac{1}{3x + 2} \).

Front

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

Back

Calculate \( g(-1) = \frac{1}{(-1)^2} = 1 \) and then \( f(1) = 1^2 = 1 \). Thus, \( (f \circ g)(-1) = 1 \).

Front

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

Back

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 composition \( g(f(x)) \) if \( f(x) = 2x \) and \( g(x) = 2x^2 - 1 \)?

Back

Substituting \( f(x) \) into \( g(x) \): \( g(f(x)) = g(2x) = 2(2x)^2 - 1 = 8x^2 - 1 \).

Front

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

Back

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

Front

What is the domain of the composition of functions?

Back

The domain of \( f \circ g \) is the set of all \( x \) in the domain of \( g \) such that \( g(x) \) is in the domain of \( f \).

Front

Define the term 'inverse function'.

Back

An inverse function reverses the effect of the original function. If \( f(x) \) is a function, its inverse is denoted as \( f^{-1}(x) \) such that \( f(f^{-1}(x)) = x \).

Front

What is the relationship between a function and its inverse?

Back

A function and its inverse are reflections of each other across the line \( y = x \).