Unit 1: Functions and Their Graphs
Relations that qualify as functions, their domains and ranges, and how shifting, stretching, combining, and inverting a graph works.
16 topics · 60 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 Is a Function
- How it worksHow does the vertical line test decide whether a graph represents a function?
- Worked exampleFind whether the relation x2 + y2 = 9 defines y as a function of x.
- Common mistakeWhy isn't every equation relating x and y automatically a function?
Your turn1 of 2
The graph of x2 + y2 = 72 is a circle, which fails the vertical line test. At x = 0 there are two y-values, 7 and -7. Enter the positive one.Domain and Range
- How it worksHow do you read a function's domain and range straight off its graph?
- Worked exampleFind the domain of f(x) = √(x - 3) / (x - 5).
- Worked exampleFind the range of f(x) = -(x - 2)2 + 5.
- Common mistakeWhy isn't the domain of a function the same thing as its range?
Your turn1 of 2
Find the smallest x-value in the domain of f(x) = √(x - 5) / (x - 11).Evaluating and Building Functions
- How it worksHow do you evaluate a function at an expression like f(x+2) instead of a number?
- Worked exampleFind f(2) and f(x+h) for f(x) = x2 - 3x + 1.
- Common mistakeWhy isn't f(x+h) the same as f(x) + f(h)?
- Worked exampleFind the difference quotient (f(x+h) - f(x))/h for f(x) = x2 + 1.
Your turn1 of 2
Let f(x) = x2 - 4x + 8. Find f(5).Piecewise-Defined Functions
- How it worksHow do you decide which piece of a piecewise function to use for a given input?
- Worked exampleFind g(-2), g(1), and g(4) for g(x) = x+5 if x<0, x2 if 0≤x<3, and 2x-1 if x≥3.
- Common mistakeWhy doesn't a piecewise function have to be discontinuous where its pieces meet?
- Common mistakeWhy isn't a piecewise function actually three separate functions instead of one function with three rules?
Your turn1 of 2
For g(x) = x + 7 if x < 0, x2 if 0 ≤ x < 3, and 5x - 1 if x ≥ 3, find g(-5).Increasing, Decreasing, and Constant Intervals
- How it worksHow do you find the intervals where a function is increasing by reading left to right along its graph?
- Worked exampleFind the intervals of increase and decrease for f(x) = x3 - 3x.
- Common mistakeWhy isn't a function increasing just because its output values are positive?
- Ties togetherHow does a function being increasing on an interval connect to its average rate of change staying positive there?
Your turn1 of 2
The function f(x) = x3 - 342 x switches from decreasing to increasing at a positive x-value. Find it.Even and Odd Functions
- How it worksHow do you test algebraically whether a function is even, odd, or neither?
- Worked exampleFind whether f(x) = x4 - 2x2 + 1 is even, odd, or neither.
- Worked exampleFind whether f(x) = x3 - x is even, odd, or neither.
- Common mistakeWhy doesn't an odd function have to look identical to its own reflection across the y-axis?
Your turn1 of 2
A function f is odd, and f(-7) = 6. Find f(7).Average Rate of Change
- How it worksHow does the slope of a secant line compute an average rate of change between two points?
- Worked exampleFind the average rate of change of f(x) = x2 + 1 from x = 1 to x = 4.
- Common mistakeWhy isn't the average rate of change over an interval the same as the rate of change at its midpoint?
Your turn1 of 2
Find the average rate of change of f(x) = x2 + 8 from x = 3 to x = 8.Vertical and Horizontal Shifts
- How it worksHow does replacing x with x - h in a formula move a graph horizontally?
- Worked exampleFind the equation of y = x2 after it is shifted 3 units left and 2 units down.
- Worked exampleFind the equation of y = √x after it is shifted 4 units up and 1 unit right.
- Common mistakeWhy does f(x - 3) shift a graph to the right instead of to the left?
Your turn1 of 2
The graph of y = x2 is shifted 4 units left and 6 units down. Write the equation of the new function.Reflections Across the Axes
- How it worksHow does negating f(x) versus negating x reflect a graph across a different axis?
- Worked exampleFind the equation of the reflection of f(x) = √x across the x-axis.
- Common mistakeWhy isn't reflecting across the x-axis the same transformation as reflecting across the y-axis?
- Ties togetherHow does a function being even connect to its graph looking unchanged after a reflection across the y-axis?
Your turn1 of 2
The graph of y = √(x - 4) is reflected across the x-axis. Write the equation of the reflected function.Stretches and Compressions
- How it worksHow does the constant a in a·f(x) stretch or compress a graph vertically?
- Worked exampleFind the equation of y = x2 after a vertical stretch by a factor of 4.
- Worked exampleFind the equation of y = x2 after a horizontal compression by a factor of 1/3.
- Common mistakeWhy does f(2x) compress a graph horizontally instead of stretching it?
Your turn1 of 2
The graph of y = x2 is vertically stretched by a factor of 7. Write the equation of the new function.Combining Transformations
- How it worksHow do you read a stretch, a reflection, and a shift all out of one equation like y = -2f(x-1)+3?
- Worked exampleFind the graph of y = -2(x-1)2 + 3 starting from the graph of y = x2.
- Going furtherWhat happens if you apply the same shift and stretch to a graph in the opposite order?
- Common mistakeWhy doesn't shifting first and stretching second land you on the same graph as stretching first and shifting second?
Your turn1 of 2
Starting from y = x2, the graph is stretched vertically by a factor of 5, then shifted right 4 and up 7. Write the equation of the resulting function.Adding, Subtracting, Multiplying, and Dividing Functions
- How it worksHow do you build (f+g)(x), (f-g)(x), and (f/g)(x) from two given functions?
- Worked exampleFind (f-g)(x) and (f·g)(x) for f(x) = x2 + 4 and g(x) = 5x - 1.
- Common mistakeWhy does the domain of f/g exclude every x where g(x) equals zero?
Your turn1 of 2
Let f(x) = x2 + 7 and g(x) = 5x - 6. Write (f - g)(x).Composition of Functions
- How it worksHow does f(g(x)) feed the output of one function into the input of another?
- Worked exampleFind f(g(x)) and g(f(x)) for f(x) = 2x + 1 and g(x) = x2.
- Common mistakeWhy isn't f(g(x)) generally the same as g(f(x))?
Your turn1 of 2
Let f(x) = 5x + 5 and g(x) = x2 + 5. Write f(g(x)) as a function of x.Domain of a Composite Function
- How it worksHow do you find the domain of f(g(x)) by checking both g's domain and where g(x) lands?
- Worked exampleFind the domain of the composite f(g(x)) for f(x) = √x and g(x) = x - 5.
- Common mistakeWhy isn't the domain of f(g(x)) automatically the same as the domain of g(x)?
- Ties togetherHow does finding the domain of a composite function combine square-root and denominator restrictions into one rule?
Your turn1 of 2
Let f(x) = √x and g(x) = x - 7. Find the smallest x-value in the domain of f(g(x)).One-to-One Functions and Inverses
- How it worksHow does the horizontal line test check whether a function has an inverse?
- How it worksWalk me through the algebra steps for finding the inverse of f(x) = 2x - 5.
- Worked exampleFind the inverse of f(x) = (x+3)/4.
- Common mistakeWhy isn't the inverse of a function the same thing as its reciprocal?
Your turn1 of 2
Let f(x) = (x + 7)/5. Find f−1(x).Graphing a Function and Its Inverse
- How it worksHow does swapping x and y coordinates reflect a graph across the line y = x?
- Worked exampleFind and graph the inverse of f(x) = x3 + 2 using the swap-and-solve method.
- Common mistakeWhy doesn't every function have a graph that reflects into another function?
- Ties togetherHow does reflecting a graph across the line y = x compare to reflecting it across the x-axis or y-axis?
Your turn1 of 2
Let f(x) = x3 + 4. Find f−1(x).