Graph the inverse of each one-to-one function.

Verified step by step guidance

1. Identify the given function on the graph. The blue line represents the function.
2. Determine the equation of the given function. Since it is a straight line, it can be written in the form y = mx + b, where m is the slope and b is the y-intercept.
3. Find the slope (m) of the line by selecting two points on the line and using the formula m = (y2 - y1) / (x2 - x1).
4. Find the y-intercept (b) by identifying the point where the line crosses the y-axis.
5. To find the inverse function, swap the x and y variables in the equation of the original function and solve for y. This new equation represents the inverse function.

Key Concepts

One-to-One Functions

A one-to-one function is a type of function where each output value is uniquely paired with one input value. This means that no two different inputs produce the same output. This property is essential for a function to have an inverse, as the inverse must also be a function, which requires that each output corresponds to exactly one input.

Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, then its inverse f⁻¹ takes y back to x. Graphically, the inverse of a function can be found by reflecting the graph of the original function across the line y = x, which helps visualize the relationship between the function and its inverse.

Graphing Techniques

Graphing techniques involve plotting points on a coordinate plane to represent mathematical functions visually. For one-to-one functions and their inverses, understanding how to accurately reflect points across the line y = x is crucial. This reflection allows for the correct representation of the inverse function, ensuring that the graph maintains the one-to-one property.