Find the range of f(x)=x^2-2 x-1 . Determine the values of x in the domain of f for which f(x)=2

Name: Problem 29, Chapter 1: Essentials of Precalculus
Uploaded: 2021-07-11T00:03:07-08:00
Duration: 2 min 23 s
Channel: Erika Bustos
Description: Find the range of f(x)=x^2-2 x-1 . Determine the values of x in the domain of f for which f(x)=2

step 1 We need to find the range of the given function, and then we need to determine the values of X,

Find the range of f(x)=x^2-2 x-1 . Determine the values of x...

Key Concepts

Solving Quadratic Equations

Solving quadratic equations involves finding the values of the variable that satisfy the equation when the function is set equal to a specific value. Whether through factoring, completing the square, or using the quadratic formula, this process is fundamental in many areas of algebra. Applying these techniques allows one to determine the specific x-values corresponding to a given function value.

Vertex and the Range

The vertex of a quadratic function is the point where the function reaches its minimum or maximum value, which in turn defines the range of the function. By rewriting the quadratic in vertex form via completing the square, one can directly identify the vertex and determine the set of all possible output values (the range) of the function.

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic expression into a perfect square plus a constant. This method rewrites the quadratic function into vertex form, making it easier to determine features such as the vertex and the minimum or maximum value of the function. It's a critical tool for analyzing and solving quadratic equations.

Quadratic Functions

A quadratic function is a second-degree polynomial that has the form ax² + bx + c, where a, b, and c are constants. Its graph is a parabola, which either opens upward or downward depending on the sign of the coefficient a. Understanding quadratic functions is essential because they appear in various contexts in mathematics and have predictable properties such as symmetry and a vertex representing either a minimum or maximum value.

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