Function transformations and notation — Digital SAT Math practice
Shifts, reflections, composition, piecewise and symmetry.
Key terms
- Domain: The set of all valid input values for which the function produces a real output.
- Range: The set of all possible output values produced by the function over its domain.
- Piecewise Function: A function defined by multiple sub-functions, each applying to a specific interval of inputs.
- Vertex: The minimum or maximum point where the direction of a graph changes, often at (h,k).
- Composition: Combining two functions where the output of one becomes the input of another.
Common traps
- Horizontal shifts move in the opposite direction of the sign inside the parentheses.
- Composition f(g(x)) requires evaluating g first, not plugging x into both functions simultaneously.
- Reflecting over the y-axis negates the input variable, while reflecting over the x-axis negates the entire function output.
- Piecewise functions require checking which interval actually contains your specific input value before applying a rule.
Key ideas to know
Vertical and Horizontal Shifts
If g(x) = f(x + 3) - 2, how is the graph of g related to the graph of f?
The graph of f shifts left 3 units and down 2 units. Adding inside x shifts horizontally opposite; subtracting outside shifts vertically down.
Watch out: Students often shift right for f(x + h) instead of left, or confuse the direction of horizontal shifts.
Evaluating Composite Functions
Given f(x) = 2x + 1 and g(x) = x2, what is the value of f(g(3))?
First evaluate the inner function g(3) = 9, then plug that result into f: f(9) = 2(9) + 1 = 19.
Watch out: Plugging the inner result into the wrong function or evaluating outer before inner changes the answer entirely.
Identifying Function Symmetry
How do you determine whether h(x) = x4 - 5x2 + 1 is even, odd, or neither?
Substitute -x for every x. If h(-x) equals h(x), it is even (y-axis symmetry). If h(-x) equals -h(x), it is odd (origin symmetry). Otherwise, neither.
Watch out: Checking only the constant term or graph shape instead of algebraically testing every term with -x.
Choosing the Correct Piecewise Rule
A student calculates p(2) for p(x) = { x + 1 if x < 2; 3x - 4 if x >= 2 } and gets 3 by using x + 1. What is the error?
The input x = 2 does not satisfy x < 2, so the first rule cannot be used. The correct rule is 3x - 4, giving p(2) = 3(2) - 4 = 2.
Watch out: Ignoring whether the inequality includes the equals sign and picking the wrong interval for boundary values.
Sample questions
Question 1
The graph of y = f(x) contains the point (2, -3). Which point must lie on the graph of g(x) = f(x + 5) - 4?
- (-3, -7)
- (-3, 1)
- (7, -7)
- (7, 1)
Show answer and explanation
Answer: (-3, -7)
To use f(2), set x + 5 = 2, which gives x = -3. Then g(-3) = f(2) - 4 = -3 - 4 = -7, so (-3, -7) lies on g.