Inverse Functions and Graph Transformations (Video Notes)

Vocabulary terms covering inverse functions, how they relate to graphs, composition checks, and common function transformations discussed in the video notes.

Inverse function

A function that undoes another function by swapping inputs and outputs; if f and g are inverses, then f(g(x)) = x and g(f(x)) = x.

Graph of an inverse function

The graph of the inverse is related to the original by swapping coordinates of points, effectively reflecting across the line y = x.

Outputs become inputs

A key idea for inverses: the outputs of the original function become the inputs of the inverse.

Inputs become outputs

For inverse functions, the inputs of the original function become the outputs of the inverse.

Composition of functions

Applying one function to the result of another: f(g(x)). This is central to checking if two functions are inverses.

f(g(x)) = x test for inverses

If composing f with g yields the identity function x, then f and g are inverses.

g(f(x)) = x test for inverses

If composing g with f yields the identity function x, then f and g are inverses.

Fifth root function

A radical function of the form f(x) = x^(1/5); in the notes, an example includes transforming this function by adding constants (e.g., +1, +2).

Transformation of a parent function

Viewing a new function as a transformed version of a simple, standard (parent) function, using shifts, reflections, and stretches to predict its shape.

Horizontal and vertical shifts

Shifts caused by adding/subtracting constants inside or outside the function, e.g., f(x) = root(x + 1) + 2 moves the graph left and up.

Slope / rate of change

The amount by which y changes per unit change in x; for linear functions, this is constant and called the slope.

Linear function

A function of the form y = mx + b with constant slope m; its graph is a straight line.

y-intercept

The point where the graph crosses the y-axis (0, b); indicates the vertical position of the graph.

x-intercept

The point where the graph crosses the x-axis (a, 0); found by solving f(x) = 0.

Reflection property of inverse graphs

If a point (a, b) lies on the graph of f, then (b, a) lies on the graph of the inverse, illustrating the swap of inputs and outputs.