What is a function in mathematics?

A function is a relationship between a set of inputs and a set of possible outputs, where each input is related to exactly one output. It is often written as f(x), where f is the function and x is the input.

How do you determine if a relation is a function?

To determine if a relation is a function, use the vertical line test: if a vertical line intersects the graph of the relation at more than one point, then the relation is not a function.

What is the domain of a function?

The domain of a function is the complete set of possible values of the independent variable (input) for which the function is defined. For example, in f(x) = 1/x, the domain excludes x = 0.

Evaluate the function f(x) = 2x² + 3 at x = 4.

To evaluate the function, substitute 4 for x: f(4) = 2(4)² + 3 = 2(16) + 3 = 32 + 3 = 35.

If f(x) = 3x - 5 and g(x) = 2x + 4, what is (f + g)(x)?

To find (f + g)(x), add the two functions together: (f + g)(x) = f(x) + g(x) = (3x - 5) + (2x + 4) = 5x - 1.

What is an even function?

A function f(x) is even if for every x in the domain, f(x) = f(-x). Even functions are symmetric about the y-axis.

What is an odd function?

A function f(x) is odd if for every x in the domain, f(x) = -f(-x). Odd functions are symmetric about the origin.

Identify if f(x) = x² + 3 is even, odd, or neither.

This function is even since f(-x) = (-x)² + 3 = x² + 3 = f(x).

If f(x) = 2x + 3, what is f(0)?

Substituting 0 in f(x): f(0) = 2(0) + 3 = 3.

What transformation occurs to f(x) when it changes to f(x + 2)?

The graph of the function f(x) shifts left by 2 units.

Find the value of x for which f(x) = 0 if f(x) = x² - 4.

Setting the equation to zero: x² - 4 = 0 gives x = ±2.

If f(x) = 3x² - 6x + 2, what is the vertex of the parabola?

The vertex x-coordinate is found using -b/(2a): x = 6/(2*3) = 1. Plugging back, f(1) = 3(1)² - 6(1) + 2 = -1. Hence, vertex is (1, -1).

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

The range is [0, ∞) because x² is always non-negative.

If f(x) = x³ - 3x, find f(2).

Substituting 2 into f(x): f(2) = 2³ - 3(2) = 8 - 6 = 2.

Evaluate g(x) = |x - 3| when x = 5.

Substituting 5 into g(x): g(5) = |5 - 3| = |2| = 2.