Become a Full Master of Calculus 1 - Learn step by step

Define a function as a relation from x to y via ordered pairs. Each input gives a single output, and notations f(x), g(x), and h(x) name functions.

Explore the classification of functions by type, including algebraic and polynomial functions (quadratic to fifth degree), rational and power functions, and the transcendental family with trigonometric, logarithmic, and exponential cases.

Determine if a function is even or odd by testing f(-x) against f(x) and -f(x). Use examples with even and odd powers and with sine and cosine to illustrate parity.

Visualize how graphs symmetric about the y-axis indicate even functions, while symmetry about the origin indicates odd functions, using x^2 and sin x as examples.

Explore the product of two functions, f and g, expand f(x)g(x) using distribution, and simplify polynomials with concrete examples like (x^2+x)(2x+1) and evaluating at x=2.

Learn to add and subtract functions by combining like powers, using examples f(x)=2x^2+2x and g(x)=2x+1, and a subtraction case.

Learn to identify the domain and range of a function from real numbers on the x and y axes, use inequalities, and represent endpoints with parentheses or brackets.

Identify the domain and range from parabolic graphs using end-point conventions. Use interval notation and filled circles to indicate inclusive bounds, e.g., domain [-4, 9], range [-4, 9].

Identify the domain and range from parabola graphs, using filled and empty circles to determine interval notation and vertex, and apply it to several cases.

Identify the domain and range from graphs of straight lines by reading arrowheads and end circles on the x and y axes, using inequality notation.

Identify the domain and range from a discrete graph by listing the x and y coordinates of plotted points; contrast with a continuous graph using endpoints that extend to infinity.

Identify domain and range from discrete circle-like graphs by listing unique x and y coordinates, noting discontinuities, and compare with a continuous circle’s borders on the x and y axes.

Determine the domain of polynomial functions as all real numbers. For fractional functions, exclude zeros of the denominator, e.g., x ≠ 5 and x ≠ ±3, to prevent indeterminate forms.

Identify the domain of a fractional function with a quadratic denominator by factoring and excluding its zeros; interval testing shows all real numbers except -2 and 8.

Discover how to find the domain of square root functions by ensuring the inside part is nonnegative, using examples like x+1, 3−x, and x^2−x−2.

Identify the domain of the square root of x^2 - x - 2 by factoring to (x-2)(x+1) and requiring a nonnegative product, yielding x ≤ -1 or x ≥ 2.

Find the domain by ensuring the expression under the square root is nonnegative and the denominator nonzero. Show example 14: x≥1 and x≠3; show example 15: x≥4 (x≠-1).

Use the ordered pair test to decide if a relation is a function. Ensure each domain maps to a single range; multiple ranges make it not a function.

Explore one-to-one and onto functions by analyzing how every input maps to outputs, ensuring all outputs link to inputs within the domain and range.

Analyze ordered pairs to determine if a function is onto, by comparing domain and range and distinguishing onto from one-to-one mappings.

Use the ordered pair test to decide if a relation is a function, ensuring each input maps to one output, and assess onto status.

Explore how to draw the graph of a function and translate mathematics into a two-dimensional picture on the x and y axes, for analysis in business, research, and medical sciences.

Learn the basic technique to graph any function by selecting x-values, building a value table, plotting points, and connecting them to reveal linear or parabolic shapes.

Learn how to graph a three-piece piecewise function with 2x^2+1 for x<0, x for x≥0, and 0 for x<-2, using border rules and the vertical line test.

Derive the equations for each piece of a three-piece function from its graph, including a parabola and two lines with domain borders.

Explore how adding or subtracting values shifts the graph of a function in the x y plane, using y equals x squared and y equals 2 x squared as examples.

Explore how multiplying a function by a constant and applying a negative sign transforms its graph, illustrated with y = x^2 and scaled or reflected versions.

Transform the graph of y = 2x^2 by changing the x-variable with factors 3 and -3, squeezing the graph along the x-axis and swapping left and right sides.

Multiply the cubic function by a negative, substitute x values, and plot the points to form graph; observe the reflection across the x-axis and how the equation changes the cubic.

Explore composite functions as the combination of two functions, illustrated by f∘g and g∘f, using the hollow circle symbol and noting how order affects the result.

Learn how to compute f composed with g by substituting values in diverse function pairs, including linear and quadratic functions, roots, natural logs, exponentials, and tangents, via explicit examples.

Explore the inverse function as the reverse of a relation, and learn when a one-to-one function passes the horizontal line test to possess an inverse, while noting vertical line test.

Explore graphical intuition of inverse functions by reflecting a function across the line y equals x, showing how the inverse mirrors the original and demonstrates horizontal and vertical line tests.

Learn the technique to find the inverse function by swapping x and y, turning original function values into the inverse relation, and showing how x-values become y-values.