What is a function in mathematics?

A function is a relation that assigns exactly one output for each input. It can be represented as f(x), where 'f' is the function name and 'x' is the input variable.

How can you determine if a 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) for which the function is defined. For example, in f(x) = √x, the domain is x ≥ 0 because the square root of a negative number is not defined in the real number system.

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

The range of the function f(x) = x² is the set of all possible output values (y-values). Since x² is always non-negative, the range is y ≥ 0.

If f(x) = 2x + 3, what is f(4)?

Substitute 4 for x in the function.

To find f(4), substitute 4 for x in the function: f(4) = 2(4) + 3 = 8 + 3 = 11. Therefore, f(4) = 11.

What is the formula for the composition of functions?

The composition of functions is given by (f o g)(x) = f(g(x)). For example, if f(x) = 2x and g(x) = x + 1, then (f o g)(x) = f(g(x)) = f(x + 1) = 2(x + 1) = 2x + 2.

What is an inverse function?

An inverse function undoes the operation of the original function. If f(x) is a function, its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x for all x in the domain of f⁻¹.

If f(x) = x² - 4, what are the x-intercepts of the function?

Solve for x when f(x) = 0.

To find the x-intercepts, set f(x) = 0: x² - 4 = 0. Solving gives x² = 4, hence x = ±2. The x-intercepts are x = 2 and x = -2.

What is the vertex form of a quadratic function?

The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify the vertex and the direction of the parabola.

If f(x) = 3x - 2 and g(x) = x², what is (f o g)(2)?

First find g(2) and then apply f.

First, calculate g(2): g(2) = 2² = 4. Now apply f: f(g(2)) = f(4) = 3(4) - 2 = 12 - 2 = 10. Thus, (f o g)(2) = 10.