MathLab with Mr. Ahmed

MathLab with Mr. Ahmed is a math learning website by Ahmed Amasha, a mathematics teacher teaching the American curriculum. Explore Algebra 2, Precalculus, Calculus, and SAT and PSAT Math Prep through lessons, focused practice and printable resources.

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2.2 Graphs of Functions

Identify allowed inputs and possible outputs.

Before you solve

Ideas in this lesson

Explore the lesson brief and the worked example for Idea 11. Create a free account to continue with the remaining ideas and practice.

01Connect graph points, values, and tablesWorked steps · five questions02Draw lines, constants, and restricted segmentsWorked steps · five questions03Draw quadratic and higher-power graphsWorked steps · five questions04Draw square-root and odd-root graphsWorked steps · five questions05Draw reciprocal graphs without false connectionsWorked steps · five questions06Graph absolute values and sign rulesWorked steps · five questions07Select and justify graphing windowsWorked steps · five questions08Construct piecewise graphs correctlyWorked steps · five questions09Graph floor functions and step boundariesWorked steps · five questions10Use the vertical-line test and read finite domainsWorked steps · five questions11Determine whether implicit equations define a functionWorked steps · five questions12Recover formulas from segments and semicirclesWorked steps · five questions13Compare power, root, shift, and scale familiesWorked steps · five questions14Build and graph contextual cost and physical modelsWorked steps · five questions15Investigate inverse-power relations and compressed stepsWorked steps · five questions16Construct and explain graphs of absolute outputsWorked steps · five questions
Lesson brief and rules

2.2 Graphs of Functions

The domain is the set of allowed inputs; the range is the set of outputs. Over the real numbers, the expression under a square root must be nonnegative.

Remember
For rational functions, exclude every input that makes the original denominator zero.
Worked example 11
01 / For f(x)=x2−2f(x)=x^{2} - 2, make a five-row table at x=−2,−1,0,1,2x=-2,-1,0,1,2; plot the points and sketch the full curve. What point corresponds to input 33? Write or draw your full response before checking the answer. A teacher must review the construction or argument. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
02 / The table outputs are [2,−1,−2,−1,2][2, -1, -2, -1, 2].
03 / Plot input horizontally and output vertically.
04 / The full graph is an upward parabola with vertex (0,−2)(0,-2); f(3)=7f(3)=7.
Worked example 2 · Connect graph points, values, and tables

The entire function graph is the four filled points shown. Find f(−2)f(-2) and list its domain.

Four-point functionAxes labeled x and y. Window x from -3 to 3, y from -2 to 4. Open circles exclude endpoints; filled circles include them.-3-2-10123-2-101234xy

Find the dot vertically above x=−2x=-2.
Its height is 33.
Only the four filled dots belong to this graph, so the domain contains four inputs, not the whole interval.

Read the graphs

Connect graph points, values, and tables

The entire function graph is the four filled points shown. Find f(−2)f(-2) and list its domain.
Four-point functionAxes labeled x and y. Window x from -3 to 3, y from -2 to 6. Open circles exclude endpoints; filled circles include them.-3-2-10123-2-10123456xy

Find the dot vertically above x=−2x=-2.
Its height is 55.
Only the four filled dots belong to this graph, so the domain contains four inputs, not the whole interval.

Answer: f(−2)=5; D={−2,−1,0,2}f(-2)=5;\ D=\{-2,-1,0,2\}

Use the vertical-line test and read finite domains

Apply the vertical-line test to graph 22. Its endpoints are included. If it is a function, give domain and range.
Restricted U-shaped graphAxes labeled x and y. Window x from -3 to 3, y from -2 to 3. Open circles exclude endpoints; filled circles include them.-3-2-10123-2-10123f: solidxy

Every vertical line meets the drawn curve at most once.
The horizontal span is −2-2 through 22, with endpoints.
The lowest height is −1-1 and highest is 11.

Answer: Yes; D=[−2,2], R=[−1,1].\text{Yes; }D=[-2,2],\ R=[-1,1].

Recover formulas from segments and semicircles

Find a formula and its full domain for the segment shown.
Line segment to modelAxes labeled x and y. Window x from -3 to 3, y from -1 to 5. Open circles exclude endpoints; filled circles include them.-3-2-10123-1012345f: solidxy

Calculate rise/run from the two endpoints.
Use either point to find the vertical intercept.
The filled endpoints limit the function to the closed interval [−2,2][-2,2].

Answer: f(x)=x+2, −2≤x≤2f(x)=x+2,\ -2\le x\le2

15 questions with a worked answer for each.Start practice ↗
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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