11.1 Functions: Definition, Function Notation & Evaluating Linear and Quadratic Functions

Why This Topic Matters

Assessment targets A.7.b and A.7.c live inside "Compare, represent, and evaluate functions," part of the Algebraic Problem Solving content area that makes up about 55% of the GED Mathematical Reasoning test. Function questions appear as multiple-choice items, fill-in-the-blank items, and drag-and-drop items where you identify a value, complete a table, or pick the graph that matches a rule. GED Testing Service's official Assessment Guide rates these skills at Depth of Knowledge (DOK) level 1-2 — meaning the test expects you to correctly apply a definition or substitute a value, not derive a proof. That makes this one of the highest-payoff topics to master: the mechanics are learnable in an afternoon, but a single sign error or a misread notation costs an otherwise-easy point.

What Is a Function?

A function is a relationship between two sets of numbers — called the domain (the set of allowed input values) and the range (the set of resulting output values) — in which every input produces exactly one output. If even one input value produces two different outputs, the relationship is not a function.

GED items test this idea (assessment target A.7.b) in three formats: a table of input-output pairs, a mapping diagram, or a graph. Each format has its own quick check:

Common trap: test-takers often assume any smooth, connected curve is automatically a function. A circle is smooth and connected, but a vertical line drawn through its center crosses it twice — so a circle graphed as x^2 + y^2 = r^2 is not a function of x. A standard upward- or downward-opening parabola (y = ax^2 + bx + c), by contrast, always passes the vertical line test and is a function.

Worked example (table check): Does this table represent a function?

No — the input x = 2 appears twice, once paired with y = 7 and once with y = 9. Since one input produces two different outputs, this table does not represent a function.

Function Notation: Reading and Writing f(x)

Once a relationship is confirmed to be a function, GED problems typically name it using function notation: f(x), read "f of x." This notation replaces the more familiar y in an equation like y = 2x - 5, which can also be written f(x) = 2x - 5. The letter in front (commonly f, but sometimes g or h) simply labels which function is being described — it does not represent a variable being multiplied by x.

Common trap: the single most frequent GED function error is misreading f(x) as "f times x." Remember: f(x) is a single output value — the y-value the function produces for that particular input. When you see f(3), it means "the output of function f when the input is 3," not "f times 3."

Evaluating Linear Functions

To evaluate a function at a given input, substitute that number for every x in the rule, then simplify using order of operations.

Worked example: Given f(x) = 3x - 7, find f(4).

1. Substitute 4 for x: f(4) = 3(4) - 7
2. Multiply: f(4) = 12 - 7
3. Subtract: f(4) = 5

So f(4) = 5 — meaning when the input is 4, this function's output is 5.

Worked example with a negative input: Given f(x) = -2x + 9, find f(-3).

1. Substitute -3 for x: f(-3) = -2(-3) + 9
2. Multiply (negative times negative is positive): f(-3) = 6 + 9
3. Add: f(-3) = 15

Common trap: dropping the negative sign when substituting a negative input, or forgetting to apply the multiplication sign rule (negative times negative equals positive). Always wrap a negative substitution in parentheses before multiplying.

Evaluating Quadratic Functions

Assessment target A.7.c specifically calls out evaluating linear and quadratic functions in function notation, so GED Math includes items with an x^2 term.

Worked example: Given f(x) = x^2 - 4x + 3, find f(-3).

1. Substitute -3 for every x: f(-3) = (-3)^2 - 4(-3) + 3
2. Square first: (-3)^2 = 9, so f(-3) = 9 - 4(-3) + 3
3. Multiply: -4(-3) = 12, so f(-3) = 9 + 12 + 3
4. Add: f(-3) = 24

Common trap: squaring a negative number incorrectly. (-3)^2 means (-3) x (-3) = 9, a positive result — it does not equal -9. Many test-takers apply the exponent only to the 3 and then attach a negative sign afterward, which is the single most common quadratic-evaluation error on the GED. Always square the entire signed number, parentheses and all.

Worked example (second quadratic input): Using the same function f(x) = x^2 - 4x + 3, find f(0). Substituting 0 for x gives f(0) = 0^2 - 4(0) + 3 = 0 - 0 + 3 = 3. Notice that f(0) always equals the constant term of the quadratic — a fast mental-math shortcut worth remembering for multiple-choice elimination.

Real-World Function Scenarios

GED items often frame function evaluation inside a workforce or academic context. For example: "A rental car company charges a flat $35 fee plus $0.20 per mile driven. If the cost, in dollars, is modeled by the function c(m) = 0.20m + 35, what is the cost of driving 60 miles?" Substituting m = 60 gives c(60) = 0.20(60) + 35 = 12 + 35 = $47. Treat the function name and variable (here, c and m instead of f and x) exactly the same way — the letters change, but the substitution process does not.