← Calculus I

Unit 1: Functions and Precalculus Review

The toolkit calculus assumes you already own: what a function is, how its graph moves, and the trig, exponential, and log functions in every later chapter.

15 topics · 59 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

1.1 What a Function Actually Is

  1. How it worksHow does the vertical line test turn one input, one output into something you can see?
  2. Worked exampleFind the domain and range of f(x) = 1/(x - 3) directly from its formula.
  3. Common mistakeWhy isn't every equation relating x and y automatically a function?
  4. Going furtherWhat happens to a function's graph when you restrict its domain to only positive x?
Your turn1 of 2
Let f(x) = 1/(x - 4). Evaluate f(11).

1.1 Domain and Range from a Formula and a Graph

  1. How it worksHow do you read a function's range straight off its graph without a formula?
  2. Worked exampleFind the domain of g(x) = square root of (5 - x) and write it in interval notation.
  3. Common mistakeWhy isn't the range of f(x) = x2 all real numbers?
  4. Worked exampleFind the domain of h(x) = 1/square root of (x - 2).
  5. Going furtherWhat happens to the domain of a function when you divide it by another function that has its own restrictions?
Your turn1 of 2
Find the right endpoint of the domain, written (-infinity, ?], of g(x) = square root of (12 - x).

1.1 Piecewise-Defined Functions

  1. How it worksHow do you decide which piece of a piecewise function to evaluate for a given input?
  2. Common mistakeWhy doesn't a piecewise function have to stay continuous where its pieces meet?
  3. Going furtherWhat happens to the graph when two pieces disagree at the break point?
  4. Worked exampleFind f(2) and f(5) for a piecewise function that switches formulas at x = 3.
Your turn1 of 2
A piecewise function is defined by f(x) = x2 for x < 5, and f(x) = 4x + 5 for x ≥ 5. Find f(5 - 2) + f(5 + 1).

1.1 Shifting, Stretching, and Reflecting Graphs

  1. How it worksHow do a vertical shift, a horizontal shift, and a reflection stack onto one base graph?
  2. Common mistakeWhy isn't f(x + 2) the same transformation as f(x) + 2?
  3. Worked exampleDraw the graph of y = -2(x - 1)2 + 3 by transforming y = x2 one step at a time.
  4. Going furtherWhat happens to a graph's x-intercepts when the entire function is shifted upward by a constant?
Your turn1 of 2
A function f(x) = x2 is shifted right by 1, stretched vertically by a factor of 3, and shifted up by 4. Write the resulting function g(x).

1.1 Composing Two Functions

  1. How it worksHow does f(g(x)) run g first and hand its output to f?
  2. Common mistakeWhy isn't function composition the same operation as multiplying two functions?
  3. Going furtherWhat happens to the domain of f(g(x)) when g's output lands outside f's domain?
  4. Worked exampleFind f(g(x)) and g(f(x)) for f(x) = x2 and g(x) = x + 1, and compare them.
Your turn1 of 2
Let f(x) = x2 and g(x) = x + 4. Find f(g(x)), as a function of x.

1.2 Polynomial and Rational Functions and Their End Behavior

  1. How it worksHow does the degree and leading coefficient of a polynomial decide which way its ends point?
  2. Going furtherWhat happens to a rational function's graph near an input that makes the denominator zero?
  3. Worked exampleFind the end behavior of f(x) = -3x4 + 2x - 7 as x goes to positive and negative infinity.
  4. Common mistakeWhy doesn't a rational function's end behavior depend on its numerator and denominator equally?
Your turn1 of 2
As x approaches positive and negative infinity, what is the end behavior of f(x) = -4x4 + 4x - 8?

1.3 The Unit Circle and the Six Trigonometric Functions

  1. How it worksHow does a point traveling around the unit circle generate the sine and cosine graphs side by side?
  2. Worked exampleFind sin, cos, and tan of 2·π/3 using the unit circle.
  3. Worked exampleFind sin, cos, and tan of 5·π/4 using reference angles on the unit circle.
  4. Common mistakeWhy isn't tangent defined at every angle the way sine and cosine are?
  5. Going furtherWhat happens to the six trig function values as an angle sweeps from the first quadrant into the second?
Your turn1 of 2
Using the unit circle, find sin(4·π/6) for the given integer m.

1.3 Amplitude, Period, and Phase Shift of Trig Graphs

  1. How it worksHow do amplitude, period, and phase shift reshape the basic sine graph into a specific model?
  2. Worked exampleDraw the graph of y = 3sin(2x - π) over one full period.
  3. Going furtherWhat happens to the period of sin(x) when you replace x with 4x inside it?
Your turn1 of 2
Find the period of y = 5sin(5x).

1.4 Inverse Functions and the Horizontal Line Test

  1. How it worksHow do you find a formula for the inverse of a one-to-one function algebraically?
  2. Going furtherWhat happens to a function's graph when you reflect it across the line y = x?
  3. Common mistakeWhy isn't f(x) = x2 invertible over all real numbers?
  4. Worked exampleFind the inverse of f(x) = (x + 3)/2 and check it by composing the two functions.
Your turn1 of 2
Find the inverse function f−1(x) for f(x) = (4x + 4)/5.

1.5 Exponential Functions and the Number e

  1. How it worksHow does raising a base greater than 1 to bigger and bigger powers produce runaway growth?
  2. Worked exampleFind the value of an account starting at 1000 dollars growing at a continuous rate of 5 percent after 3 years.
  3. Common mistakeWhy isn't exponential growth the same thing as an ordinary polynomial getting bigger?
  4. Going furtherWhat happens to the graph of 2x as the base is changed from 2 to 1/2?
Your turn1 of 2
An account starts at 2241 dollars and grows at a continuous rate of 0.0589 per year. Write its value A(t) as a function of the number of years t.

1.5 Logarithms as the Inverse of Exponentials

  1. How it worksHow does asking what power gives you 8 turn into the statement log base 2 of 8 equals 3?
  2. Worked exampleFind x in the equation log base 3 of x = 4 by rewriting it as an exponential.
  3. Common mistakeWhy isn't log(a + b) the same as log(a) + log(b)?
Your turn1 of 2
Let f(x) = log base 4 of x. Find its inverse function f−1(x).

1.5 Solving Equations Involving Exponentials and Logarithms

  1. How it worksHow does taking a logarithm of both sides pull a variable down out of an exponent?
  2. Worked exampleSolve the equation 32x = 20 for x using logarithms.
  3. Common mistakeWhy isn't log(x) + log(x) the same expression as log(2x)?
  4. Going furtherWhat happens to the solution of ax = b when the base a is between 0 and 1 instead of greater than 1?
  5. Worked exampleSolve the equation 5e0.2x = 40 for x.
Your turn1 of 2
Solve 6·e0.36x = 57 for x.

1.1 Even and Odd Symmetry

  1. How it worksHow do you test algebraically whether a function is even, odd, or neither by plugging in negative x?
  2. Worked exampleShow whether f(x) = x4 - 3x2 + 1 is even, odd, or neither.
  3. Going furtherWhat happens to the graph of an odd function when it is reflected across the y-axis?
  4. Common mistakeWhy isn't every function forced to be either even or odd?
Your turn1 of 2
Let f(x) = x4 - 4x2 + 4, which is an even function. Write f(-x) in simplified form.

1.1 Absolute Value Functions and Their Graphs

  1. How it worksHow does splitting |x| into two separate linear pieces explain its V-shaped graph?
  2. Common mistakeWhy isn't |x - 5| the same expression as x - 5 for every value of x?
  3. Worked exampleFind all values of x that satisfy |2x - 3| = 7.
Your turn1 of 2
Solve |4x - 7| = 9 for the LARGER solution of x.

1.2 Graphing a Rational Function Using Its Asymptotes and Holes

  1. How it worksHow do you locate a rational function's vertical asymptotes, holes, and horizontal asymptote all from one factored formula?
  2. Worked exampleSketch the graph of f(x) = (x - 2)/(x2 - 4) by identifying its hole, its vertical asymptote, and its horizontal asymptote.
  3. Common mistakeWhy doesn't every zero of a rational function's denominator produce a vertical asymptote?
Your turn1 of 2
For x not equal to 5, write (x - 5)/(x2 - 52) in simplified form, as a function of x.