Understanding Functions and Function Notation
Key Concepts
- Function notation: f(x), g(x)
- Inputs/outputs: what happens when you plug in values
- Domain and range
Skills Tested
- Evaluating expressions like f(3)
- Understanding that f(x) is a rule that assigns an output to each input
- Identifying domains (e.g., no division by zero or square roots of negatives)
Example Question
If f(x) = 3x^2 - 5, what is f(2)? A. 1 B. 7 C. Answer: 7 (Plug in: f(2) = 3(2)^2 - 5 = 12 - 5 = 7)
Composing and Combining Functions
Key Concepts
- Function composition: f(g(x)), g(f(x))
- Adding, subtracting, multiplying, dividing functions
Skills Tested
- Substituting one function into another
- Performing operations: (f + g)(x), (f \cdot g)(x)
Example Question
Let f(x) = 2x + 1, and g(x) = x^2. What is f(g(2))? A. 9 B. 11 C. Answer: 9 (First g(2) = 4, then f(4) = 2(4) + 1 = 9)
Translating and Transforming Functions
Key Concepts
- Horizontal and vertical shifts, stretches, reflections
- From basic parent functions like y = x, 2y = x^2, y = |x|
Skills Tested
- Understanding how equations change graph shapes
- Recognizing transformations in function equations
Example Question
What transformation is applied to f(x) = x^2 to get f(x) = (x - 3)^2 + 2? A. Left 3, up 2 B. Right 3, up 2 C. Right 2, up 3 (Shift right = subtract in x; shift up = add outside)
Interpreting Graphs of Functions
Key Concepts
- Reading function values from a graph
- Identifying max/min values, intercepts, intervals
Skills Tested
- Finding f(x) or x from a graph
- Interpreting increasing/decreasing intervals
Example Question
A graph of f(x) passes through (2, 5). What is f(2)? A. 2 B. 5 C. Cannot be determined (Function value at x = 2 is the y-coordinate of the point)
Piecewise and Step Functions
Key Concepts
- Piecewise-defined functions (functions with different rules for different intervals)
- Step functions (e.g., greatest integer function)
Skills Tested
- Determining which piece of a function applies at a given input
- Evaluating function values based on input ranges
Example Question
Let f(x) = \begin{cases} x + 1 & \text{if } x < 0 \ x^2 & \text{if } x \geq 0 \end{cases} What is f(-2)? A. 3 B. 1 C. Answer: -1 (Since x = -2 < 0, use x + 1 = -2 + 1 = -1)