Ace Advanced Functions/Precalculus in 6 Hr (Complete Course)

Explore the definition of a function as a mapping from input to a single output, using x squared and the domain and range to show independent and dependent variables.

Explore the domain and range of a function, identifying domain as possible inputs and range as possible outputs, using f(x)=sqrt(x-1) to show x≥1 and y≥0.

Explore how functions appear as graphs of ordered pairs, verify one output per input with the vertical line test, and assess 1-to-1 status via the horizontal line test.

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

Learn average rate of change as the slope of the line through (a, f(a)) and (b, f(b)). See examples where the rate is 1 on [1,3] and 0 on [-3,3].

Review factoring techniques to solve equations by factoring out common factors, factoring quadratics into (x-m)(x-n) or using the quadratic formula, and applying difference of squares.

This example shows behavior with roots at -2, 1, and 3; a horizontal asymptote at -2; a local min at -1 and max at 2; inflection near 2.5 and 2.4.

Compute the average rate of change for f(x)=x^2−x+1 on [0,3] and [−1,1], using delta y over delta x, and illustrate the slopes graphically.

Explore a piecewise function defined by x^2 for x ≤ -1 and 2x for x > -1, observe sample values, and sketch the graph with the boundary at -1.

Examine multivariable functions and vector outputs through examples like P: R -> R^3 and Q: R^2 -> R^2, showing how inputs map to a single output.

Explore treating functions as objects; form constant multiples, sums, products, and quotients for two functions, respect domain rules, and learn composition f∘g, not commutative.

Explore transformations of functions, including translations (vertical and horizontal), vertical and horizontal stretching or compression, and reflections about the x-axis and y-axis, using f(x) and constants.

Determine if a function is even or odd by evaluating f(-x); even functions are preserved under y-axis reflection, odd functions under 180-degree rotation, illustrated by 2x^2, 2x^3, and 2x^2+x.

Learn inverse functions and function composition, illustrate with F(x)=x^3 and G(x)=cube root x, and explain invertibility as one-to-one via the horizontal line test.

Explore the properties of inverses, including domain and range swap, the inverse relation condition F(A)=B, and graph reflection across y=x, with examples using sqrt(x+1) and domain restrictions.

Identify the domains of F(x)=1/x and G(x)=square root of x plus one, and derive domains for their combinations. Explore expressions for 3F, F+G, F/G, F∘G, and G∘F with input restrictions.

Explore transformations of functions, including translations, stretches, compressions, and reflections. Use F(x)=3+2x−x^2 to analyze vertical and horizontal shifts, and compute new coordinates.

Identify whether a function is even, odd, or neither by computing f(-x) and comparing to f(x) or -f(x); examples include 1/x (odd), x^4 - x^2 (even), and (2+x)/(1+x^2) (neither).

Explore inverse functions through two examples, learn why a graph may fail the horizontal line test, and restrict domains to obtain inverses; derive and verify algebraic inverses.

Learn how algebraic functions are built from constants, variables, and arithmetic operations, including power, polynomial, and rational functions, with examples and discussions of domains and graphs.

Study power functions, expressed as A x^K with rational K. Distinguish them from non power cases and see how eight graph types arise from even and odd exponents.

Study polynomial functions as finite sums of power terms; identify coefficients, leading coefficient, and leading term; analyze end behavior and turning points to infer roots and graph shape.

Explore rational functions, defined as quotients of polynomials, and learn to identify domains, vertical and horizontal asymptotes, holes, and how degrees and multiplicities affect graph behavior.

Explore the absolute value function as a piecewise graph of x and negative x, and learn how |g(x)| equals g(x) or -g(x) depending on sign, with examples.

Learn how to determine domains of power functions by rewriting to expose roots and denominators, and applying interval notation to F, G, H, and K.

Analyze a polynomial graph to determine degree, leading coefficient, and roots; identify a quartic with a repeated root at x=2 and y-intercept 6, giving F(x)=1/2(x+3)(x+1)(x-2)^2.

From P(x) and Q(x), identify a root at -2, a hole at 1, vertical asymptotes at -1 and 2, and a horizontal asymptote at y = 2, with y-intercept (0,2).

Rewrite the absolute value function f(x)=|x^2-1| as a piecewise function, evaluate at x=-2, 0, and 1, and sketch the combined graph showing the two pieces on their respective intervals.

Learn to factor polynomials into linear factors or irreducible quadratics and use long division to obtain f(x)=d(x)q(x)+r(x). Work through examples, find quotients and remainders, and identify factors like x-2.

Apply the remainder theorem to a cubic polynomial by performing long division by x minus p and verify r = f(p) with divisors x minus two and x plus one.

apply the remainder theorem to f(x)=x^3+3x^2-kx+10, using f(5)=15 to solve for k. see how long division and the remainder theorem verify the result and discuss divisor types.

Explain how the factor theorem follows from the remainder theorem, showing that x minus p is a factor of f(x) iff f(p) equals zero, with examples using long division.

Use the factor theorem to factor polynomials with leading coefficient one by testing divisors of the constant term, then apply long division to obtain linear and irreducible quadratic factors.

Explores transcendental functions, introducing exponential, logarithmic, trigonometric, and inverse trigonometric functions, their definitions and properties, and their inverse relationships with graphical behavior.

Explore exponential functions defined as a times b^x. Base b is positive and not 1; include natural base e, growth and decay, and the zero horizontal asymptote.

Explore the logarithmic function as the inverse of exponential functions, with base B and input X, and learn key log rules, change-of-base, and the natural log special case.

Learn how the unit circle in standard position defines angles, uses radians, and converts between degrees and radians with key values like pi over four and pi over six.

Define sine, cosine, and tangent from right triangles and the unit circle, linking all six trig functions via sohcahtoa. Highlight key identities, symmetry, and the unit circle graphs with periodicity.

Construct inverse trigonometric functions by restricting domains of sine, tangent, and secant to obtain one-to-one mappings. Learn inverse notations and how radians define the inverse graphs.

Solve transcendental functions by hand using log properties and exponential transformations, and apply unit circle techniques to cosine at pi over six and inverse sine and inverse secant.

Explore how to graph transcendental functions by analyzing transformations of exponential, reciprocal, and secant functions, including horizontal and vertical translations, reflections about the x-axis, and domain considerations.

Derive a right triangle from the inverse sine of x to show sine theta equals x, then compute cosine theta as sqrt(1 - x^2).

Celebrate finishing the ace advanced functions/precalculus in 6 hr course, and leave a review to help others while providing feedback to improve future study.