Determine whether each function graphed or defined is one-to-one. y = 5|x+2|

Key Concepts

One-to-One Function

A one-to-one function is a type of function where each output value is associated with exactly one input value. This means that no two different inputs produce the same output. To determine if a function is one-to-one, one can use the horizontal line test: if any horizontal line intersects the graph of the function more than once, the function is not one-to-one.

Absolute Value Function

An absolute value function is defined as f(x) = |x|, which outputs the non-negative value of x. The graph of an absolute value function is V-shaped, with a vertex at the origin. In the case of the function y = 5|x + 2|, the graph is vertically stretched by a factor of 5 and shifted left by 2 units, affecting its symmetry and one-to-one nature.

Graphical Analysis

Graphical analysis involves examining the visual representation of a function to understand its properties, such as continuity, intercepts, and whether it is one-to-one. By analyzing the shape and behavior of the graph, one can apply tests like the horizontal line test to determine if the function meets specific criteria, such as being one-to-one.