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Introduction
Evaluating Functions helps you connect a mathematical rule to a result you can explain and check. Evaluate function rules at positive and negative domain elements. Keep negative inputs grouped during multiplication and powers. This lesson belongs to coordinate geometry and graphing and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in evaluating functions.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Use signed arithmetic and order of operations.
Previous lesson: Function Notation
Key mathematical vocabulary
- variable
- A letter representing an unknown or changing quantity.
- expression
- A mathematical phrase representing a value.
- equation
- A statement that two expressions are equal.
- function
- A rule assigning one output to each permitted input.
- domain
- The set of permitted inputs.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Substitute before calculating
Evaluate function rules at positive and negative domain elements. Keep negative inputs grouped during multiplication and powers.
Evaluating means finding the numerical output of an expression for a specified input. Replace every occurrence of a variable with the given number. Use parentheses around negative inputs, then follow the order of operations.
Function notation f(t) names the output of function f at input t; it does not mean f multiplied by t. The input must belong to the domain. Distinguish evaluating an expression from solving an equation: evaluation starts with the input, while solving seeks an input.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. Replace every t with (5): 12(5) + 23.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. Substitute the same input into the original expression to check the arithmetic.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
Evaluate 12t + 23 when t = 5.
- Replace every t with (5): 12(5) + 23.
- Multiply before adding: 60 + 23.
- The value is 83.
Answer: 83
Verify: Substitute the same input into the original expression to check the arithmetic.
Avoid this mistake: Replacing only one occurrence: substitute every matching variable.
Example 2 · Basic
Evaluate 13t + 23 when t = 5.
- Replace every t with (5): 13(5) + 23.
- Multiply before adding: 65 + 23.
- The value is 88.
Answer: 88
Verify: Substitute the same input into the original expression to check the arithmetic.
Avoid this mistake: Losing the sign when squaring: (−3)² is 9, whereas −3² is −9.
Example 3 · Intermediate
Evaluate 14t + 23 when t = -5.
- Replace every t with (-5): 14(-5) + 23.
- Multiply before adding: -70 + 23.
- The value is -47.
Answer: -47
Verify: Substitute the same input into the original expression to check the arithmetic.
Avoid this mistake: Reading f(x) as multiplication: f is the rule name.
Example 4 · Advanced
Evaluate 2t + 24 when t = -5.
- Replace every t with (-5): 2(-5) + 24.
- Multiply before adding: -10 + 24.
- The value is 14.
Answer: 14
Verify: Substitute the same input into the original expression to check the arithmetic.
Avoid this mistake: Replacing only one occurrence: substitute every matching variable.
Example 5 · Challenge
Evaluate 3t + 24 when t = -5.
- Replace every t with (-5): 3(-5) + 24.
- Multiply before adding: -15 + 24.
- The value is 9.
Answer: 9
Verify: Substitute the same input into the original expression to check the arithmetic.
Avoid this mistake: Losing the sign when squaring: (−3)² is 9, whereas −3² is −9.
Common mistakes and corrections
- Replacing only one occurrence: substitute every matching variable.
- Losing the sign when squaring: (−3)² is 9, whereas −3² is −9.
- Reading f(x) as multiplication: f is the rule name.
Real-world applications
- Calculate a delivery price from package weight. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Use a distance formula to predict a trip length at a given time. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Evaluate function rules at positive and negative domain elements. Keep negative inputs grouped during multiplication and powers.
Evaluating means finding the numerical output of an expression for a specified input. Replace every occurrence of a variable with the given number. Use parentheses around negative inputs, then follow the order of operations.
Frequently asked questions
What should I know before evaluating functions?
Use signed arithmetic and order of operations.
How can I check my work?
Substitute the same input into the original expression to check the arithmetic. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Replacing only one occurrence: substitute every matching variable. Losing the sign when squaring: (−3)² is 9, whereas −3² is −9. Reading f(x) as multiplication: f is the rule name.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Calculate a delivery price from package weight.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Evaluate function rules at positive and negative domain elements. Keep negative inputs grouped during multiplication and powers.
- Replacing only one occurrence: substitute every matching variable.
- Losing the sign when squaring: (−3)² is 9, whereas −3² is −9.
- Reading f(x) as multiplication: f is the rule name.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
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Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials