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 \( g(1) \) if \( g(x) = 4x - 3 \)?

Back

To find \( g(1) \), substitute 1 into the function: \( g(1) = 4(1) - 3 = 1 \).

Front

Calculate \( f(2) \) if \( f(x) = x^2 - 1 \).

Back

To find \( f(2) \), substitute 2 into the function: \( f(2) = 2^2 - 1 = 3 \).

Front

What is the result of \( [f \circ g](4) \) if \( f(x) = 2x^2 + 3x \) and \( g(x) = x - 1 \)?

Back

First, find \( g(4) = 4 - 1 = 3 \). Then, find \( f(3) = 2(3^2) + 3(3) = 18 + 9 = 27 \).

Front

How do you find \( f[g(h(24))] \) if \( f(x) = \frac{6}{7}x + 3 \), \( g(x) = \frac{2}{3}x - 1 \), and \( h(x) = \frac{1}{4}x + 6 \)?

Back

First, calculate \( h(24) = \frac{1}{4}(24) + 6 = 12 + 6 = 18 \). Then, \( g(18) = \frac{2}{3}(18) - 1 = 12 - 1 = 11 \). Finally, \( f(11) = \frac{6}{7}(11) + 3 = \frac{66}{7} + 3 = \frac{66}{7} + \frac{21}{7} = \frac{87}{7} \approx 12.43 \).

Front

What is the value of \( f[g(h(3))] \) if \( f(x) = x^2 - 2 \), \( g(x) = x + 9 \), and \( h(x) = -3x \)?

Back

First, calculate \( h(3) = -3(3) = -9 \). Then, \( g(-9) = -9 + 9 = 0 \). Finally, \( f(0) = 0^2 - 2 = -2 \).

Front

Explain the significance of the order in function composition.

Back

The order of composition matters because \( f(g(x)) \) is generally not the same as \( g(f(x)) \). The output of the first function becomes the input of the second.

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 'function'.

Back

A function is a relation that assigns exactly one output for each input from a set called the domain.

Front

What is the range of a function?

Back

The range of a function is the set of all possible output values (y-values) that the function can produce.