Function Composition

Front

What is function composition?

Back

Function composition is the process of applying one function to the results of another function. If f and g are two functions, the composition of f and g is denoted as (f ∘ g)(x) = f(g(x)).

Front

How do you find p(p(x)) for a function p(x)?

Back

To find p(p(x)), you substitute p(x) into itself. For example, if p(x) = 3x + 4, then p(p(x)) = p(3x + 4) = 3(3x + 4) + 4.

Front

What is the inverse of a function?

Back

The inverse of a function f(x) is a function that reverses the effect of f. If f(x) takes an input x and produces an output y, then the inverse function f^{-1}(y) takes y back to x.

Front

How do you find the inverse of the function f(x) = (1/2)x + 8?

Back

To find the inverse, swap x and y and solve for y: y = (1/2)x + 8 becomes x = (1/2)y + 8. Solving for y gives f^{-1}(x) = 2x - 16.

Front

What is the notation for function composition?

Back

Function composition is denoted by the symbol '∘'. For functions f and g, the composition is written as (f ∘ g)(x) = f(g(x)).

Front

How do you evaluate f(g(x))?

Back

To evaluate f(g(x)), first find g(x) and then substitute that result into f. For example, if f(x) = x^2 and g(x) = x - 3, then f(g(x)) = f(x - 3) = (x - 3)^2.

Front

What is the result of p(q(p(-3))) if p(x) = 3x + 4 and q(x) = 2x^2?

Back

First, calculate p(-3) = 3(-3) + 4 = -5. Then find q(-5) = 2(-5)^2 = 50. Finally, calculate p(50) = 3(50) + 4 = 154.

Front

What is the output of f(h(x)) if f(x) = x^2 - 4 and h(x) = 3x + 3?

Back

To find f(h(x)), substitute h(x) into f: f(h(x)) = f(3x + 3) = (3x + 3)^2 - 4 = 9x^2 + 18x + 9 - 4 = 9x^2 + 18x + 5.

Front

What is the domain of a function?

Back

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

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.