Use and interpret function notation, evaluate functions, identify domain and range, and read key features (intercepts, intervals of increase and decrease, maximum and minimum) from a graph or table.

What this topic is asking

The Functions category begins with the F-IF standards: Interpreting Functions. The Grade 10 MCAS expects you to use function notation fluently, to evaluate a function, to state its domain and range, and to read key features (intercepts, where it increases or decreases, maxima and minima) from a graph or table. These ideas are the shared language for every function type that follows, so getting the vocabulary exact pays off across the whole category.

Function notation and evaluation

The notation $f(x)$ is read "f of x" and names the output of the function $f$ for the input $x$. It is not multiplication. To evaluate, substitute the given input everywhere $x$ appears.

For $f(x) = 2x^2 - x + 1$, the value $f(3) = 2(3)^2 - 3 + 1 = 18 - 3 + 1 = 16$. The notation also runs in reverse: "solve $f(x) = 16$" asks which inputs give the output 16. Reading the question carefully (are you given the input or the output?) is half the battle.

Function notation lets a single statement carry a lot: $f(0) = 5$ says the output is 5 when the input is 0, which on a graph is the y-intercept $(0, 5)$.

Domain and range

The domain is every input the function accepts; the range is every output it produces. For a graph, the domain is read along the x-axis (how far left and right the graph extends) and the range along the y-axis (how far down and up).

For most polynomials the domain is all real numbers. Restrictions arise from context (a length cannot be negative) or from the type of function (you cannot divide by zero, or take an even root of a negative). A parabola opening upward with vertex $(1, -4)$ has range $y \geq -4$, because the function never goes below its minimum.

Reading key features

A graph displays several features the MCAS asks you to name:

• Intercepts. The y-intercept is where the graph meets the y-axis, at $x = 0$. An x-intercept (a zero or root) is where it meets the x-axis, where $f(x) = 0$.
• Increasing and decreasing. A function increases on an interval where the graph rises as you move left to right, and decreases where it falls. These are described as intervals of $x$.
• Maximum and minimum. A high point is a maximum, a low point is a minimum. A parabola has one (the vertex); a wavier graph may have several local ones.

The vertical line test

A relation is a function only if every input has exactly one output. On a graph, this is the vertical line test: if any vertical line crosses the graph more than once, it is not a function. A parabola passes (one output per input), but a full circle fails (two outputs for most inputs). The MCAS sometimes shows a graph or table and asks whether it represents a function; the test answers it.

Try this

Q1. If $h(x) = 5 - 2x$, find $h(4)$.

• Cue. $5 - 2(4) = -3$.

Q2. A graph has its lowest point at $(2, -3)$ and rises on both sides. What is the range?

• Cue. $y \geq -3$.