Introduction to Advanced Functions: Functions and Graphs

Define a function as a machine that assigns each input x from domain A exactly one output f(x) in set B, and relate domain and range to inputs and outputs.

Identify the domain and range of a function by describing inputs x where f(x) is defined and outputs y that occur. For example, f(x)=sqrt(x-1) has domain x≥1 and range y≥0.

Explore how to verify a function with graphs by using the vertical line test to ensure one output per input, and apply the horizontal line test to identify one-to-one functions.

Explore the properties of graphs, including roots, y-intercepts, positivity on intervals, increasing and decreasing behavior, local and global maxima and minima, concavity and inflection points, and horizontal and vertical asymptotes.

Define the average rate of change as the slope between (A, f(A)) and (B, f(B)). Compute it as [f(B)-f(A)]/[B-A], with examples showing slope 1 on [1,3] and 0 on [-3,3].

Review factoring techniques to solve equations by factoring out common factors, factoring quadratics, and applying the quadratic formula, including difference of squares, to identify zeros.

Explore domain and range through multiple function examples, evaluate compositions, determine function status and one-to-one properties, and derive domains and ranges from given expressions.

Analyze a function with roots at -2, 1, and 3, a horizontal asymptote at -2, a min at -1, and a max at 2; examine increasing, decreasing, and inflection points.

Compute the average rate of change of F(x)=x^2−x+1 on [0,3] and [−1,1], show the slopes 2 and −1 with secant lines and a graphical illustration.

Explore a piecewise function defined by x^2 for x < -1 and 2x for x > -1, with examples f(-5)=25, f(-1)=1, and f(3)=6.

Explore multivariable functions with multiple inputs and outputs, describing points in space via coordinates. Analyze P: R to R^3 and Q: R^2 to R^2, including P(2)=(6,3,2) and Q(3,2)=(1,-9).