SAT Math
SAT function questions: notation, graphs, and meaning
Evaluate SAT functions, distinguish inputs from outputs, read intercepts and transformations, and interpret linear and exponential models.
asks for the output at input . An equation such as asks which inputs produce that output. Read that distinction before substituting or graphing.
An input is not an output
If , then . Replace x everywhere with 5 and calculate . But solving means solving , which gives .
The same notation can therefore lead to substitution or an equation. Circle what is inside the parentheses and what is after the equals sign. That tells you whether the input is supplied or still unknown.
In , use parentheses for a negative input: . Typing without parentheses can square 2 and then apply the minus sign, producing a different value.
Read function values from a table or graph
A table that lists input 3 and output 11 tells you . It does not tell you . Reversing input and output changes the question unless the function's inverse is explicitly involved.
On a graph, is the y-coordinate where the x-coordinate is 3. To solve , find x-intercepts. To solve , find intersections with the horizontal line .
There may be more than one input with that output. For , both and give 4. A function assigns one output to each input; it can assign the same output to different inputs.
Linear Functions · Easy
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Respect the function’s allowed inputs
A function can have a restricted domain. For with real outputs, inputs must satisfy . For , the input 3 is excluded because it makes the denominator zero.
A story can narrow the domain further. If x counts tickets sold and the venue has 100 seats, x is an integer from 0 through 100. A graph may display values outside that interval, but those points don't belong to the stated situation. Check the algebraic and contextual restrictions before reporting a solution.
Separate a parameter from a changing input
In , m and b are fixed values defining the particular line. The input x changes. If and , the slope is .
Substitute one point: , so . The function is , and . Check the other input: .
For a context such as , the constant is the starting cost and the coefficient is dollars per hour. Explain parameters with units. "The slope is 8" is less useful than "each additional hour adds 8 dollars to the cost."
Read transformations inside and outside the parentheses
If , then has vertex . The input expression shifts the graph right 3, and adding 2 shifts it up 2.
You can check the horizontal shift by asking when the squared part becomes zero. That happens at , not . This check is more reliable than memorizing a sign rule without understanding it.
Multiplying the whole output, as in , doubles every y-coordinate. Replacing the input, as in , can have a different effect: for this function it produces . Substitute before deciding two transformed expressions are equivalent.
Work from the inside for nested functions
Let and . Then starts with and continues with .
The reversed expression gives , then . The order matters. Write the intermediate output if it keeps the work clear.
For symbolic substitution, . Replace the entire input to f with using parentheses. This is useful notation practice; don't spend all your function review on composites at the expense of interpreting the models and graphs you actually miss.
Interpret an exponential model by its time step
In , the starting value is 500 and each one-unit increase in t multiplies it by 1.04, a 4% increase. If t is years, that is a yearly change.
For with t in years, the 4% increase happens every two years. The yearly multiplier would be , rather than 1.02 exactly. Read the exponent before assigning a rate to the model.
Practice linear functions and nonlinear functions in the Math bank. For every answer, say whether you found an input, an output, or a parameter. If you cannot name it, reread the prompt before moving on.
Try it yourself
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