What is a function in mathematics?

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. For example, f(x) = 2x + 3 is a function that takes an input x, multiplies it by 2, and then adds 3.

How do you determine if a relation is a function?

To determine if a relation is a function, check if any input (x-value) is associated with more than one output (y-value). If every x-value corresponds to exactly one y-value, the relation is a function.

What is the domain of a function?

The domain of a function is the set of all possible input values (x-values) that the function can accept. For example, for the function f(x) = 1/x, the domain excludes x = 0 because division by zero is undefined.

Evaluate f(x) = 3x - 5 for x = 4. What is f(4)?

Substitute x with 4: f(4) = 3(4) - 5 = 12 - 5 = 7. Therefore, f(4) = 7.

If f(x) = x² and g(x) = 2x + 1, what is (f ∘ g)(x)?

To find (f ∘ g)(x), substitute g(x) into f. So, f(g(x)) = f(2x + 1) = (2x + 1)² = 4x² + 4x + 1. Thus, (f ∘ g)(x) = 4x² + 4x + 1.

What is the range of the function f(x) = x²?

The range of the function f(x) = x² is all non-negative real numbers, represented as [0, ∞), since squaring any real number cannot produce a negative output.

How do you find the inverse of a function?

To find the inverse of a function, switch the input and output in the equation and solve for the new output. For example, to find the inverse of f(x) = 2x + 3, set y = 2x + 3 and solve for x: x = (y - 3)/2, giving us f⁻¹(y) = (y - 3)/2.

Given the function f(x) = 5 - 2x, what is f(0)?

Substituting x = 0 into the function gives f(0) = 5 - 2(0) = 5. Therefore, f(0) = 5.

If h(x) = 3x + 4, what is the value of h(-2)?

Substituting x = -2 into the function gives h(-2) = 3(-2) + 4 = -6 + 4 = -2. Therefore, h(-2) = -2.

What is the composition of functions f(x) = x + 2 and g(x) = 3x?

The composition (f ∘ g)(x) is found by substituting g(x) into f: f(g(x)) = f(3x) = 3x + 2. Therefore, (f ∘ g)(x) = 3x + 2.