← Algebra I

Unit 7: Functions

A function is a rule with exactly one output for every input -- the same idea whether it's written as a table, a graph, or f(x).

6 topics · 22 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.

What Makes a Relation a Function

  1. How it worksHow does the vertical line test check a graph for whether it's a function?
  2. Worked exampleShow me why the graph of a circle fails the vertical line test.
  3. Common mistakeWhy doesn't a function require every output to come from only one input?
  4. Worked exampleShow me why the relation with pairs (1, 2), (1, 5), and (3, 4) is not a function.
Your turn1 of 2
A relation contains the pairs (7, 6) and (7, 19), where 6 does not equal 19. Is this relation a function?

Function Notation

  1. How it worksHow does the letter inside the parentheses tell you which value to substitute into the rule?
  2. Common mistakeWhy doesn't f(x) mean f multiplied by x?
  3. Worked exampleFind f(3) if f(x) = 2x2 - 5.
  4. Going furtherWhat changes about function notation when the input itself is an expression like f(a + 2)?
Your turn1 of 2
Find f(6) if f(x) = 5x2 - 6.

Evaluating Functions

  1. How it worksHow do you evaluate a function at an input that's itself an expression, like h(x + 1)?
  2. Worked exampleFind g(-2) if g(x) = -x2 + 3x + 1.
  3. Worked exampleFind h(4) minus h(1) if h(x) = 3x - 2.
Your turn1 of 2
Find g(-1) if g(x) = -x2 + 5x + 6.

Domain and Range

  1. How it worksHow do you find a function's domain by looking for values that would break the rule?
  2. Worked exampleFind the domain of the function f(x) = 1 divided by the quantity x minus 4.
  3. Common mistakeWhy doesn't a function's range have to include every real number just because its domain does?
  4. Going furtherWhat happens to a function's domain when a square root sits over the entire expression?
Your turn1 of 2
Find the value of x that must be excluded from the domain of f(x) = 1 divided by the quantity x minus 7.

Interpreting Graphs of Functions

  1. How it worksHow do you find a function's domain, range, and intercepts just by looking at its graph?
  2. Worked exampleShow me how to read the value of f(2) directly off a graphed function.
  3. Going furtherWhat happens to a function's graph at the exact points where it switches from increasing to decreasing?
  4. Common mistakeWhy doesn't a graph that touches the x-axis mean the function's value is undefined there?
Your turn1 of 2
A graph shows a function crossing the x-axis at x = 2 without stopping there -- the curve continues on both sides. What is true about the function's value at x = 2?

Comparing Functions in Different Forms

  1. How it worksHow do you convert a function shown as a graph into a table so it can be compared to an equation?
  2. Worked exampleCompare the rate of change of a function given as a table against y = 3x + 2 given as an equation.
  3. Ties togetherHow does comparing two linear functions' slopes connect back to comparing steepness on a graph?
Your turn1 of 2
A table shows a linear function passing through (2, 6) and (6, 18). Find its rate of change and compare it to the slope of y = 2x + 3: give the table function's rate of change.