Linear and Quadratic Inverses

Front

What is the definition of an inverse function?

Back

An inverse function reverses the operation of the original function. If the function f takes an input x and produces an output y, then the inverse function f⁻¹ takes y as input and produces x as output.

Front

How do you find the inverse of a linear function?

Back

To find the inverse of a linear function f(x) = mx + b, swap x and y, then solve for y. The inverse is f⁻¹(x) = (x - b)/m.

Front

What is the inverse of the function f(x) = -4x - 12?

Back

f⁻¹(x) = -1/4x - 3.

Front

What is the general form of a quadratic function?

Back

A quadratic function is generally expressed as f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0.

Front

What is the inverse of the function f(x) = x² - 5?

Back

f⁻¹(x) = ±√(x + 5).

Front

How do you find the inverse of a function graphically?

Back

To find the inverse graphically, reflect the graph of the function across the line y = x.

Front

What is the domain of the inverse function of f(x) = x² - 5?

Back

The domain of the inverse function f⁻¹(x) = ±√(x + 5) is x ≥ -5.

Front

What is the range of the function f(x) = -4x - 12?

Back

The range of f(x) = -4x - 12 is all real numbers, as it is a linear function.

Front

What is the process to find the inverse of f(x) = (x - 2)/6?

Back

To find the inverse, swap x and y: x = (y - 2)/6, then solve for y to get f⁻¹(x) = 6x + 2.

Front

What is the significance of the line y = x in relation to inverse functions?

Back

The line y = x serves as a mirror line; a function and its inverse are symmetric with respect to this line.