Functions as a Unifying Language: cumulative review

Exit check: solve and verify without referring to the displayed steps. For $g(x) = x^{3} - 1,$ describe domain, range, intercepts, and end behavior.

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Classify the mathematical object and requested action in this lesson case: Evaluate $f(-2), f(1),$ and $f(4)$ when $f(x) = x + 3$ for $x < 1, f(x) = x^{2}$ for $1 \le x \le 3,$ and $f(x) = 10 - x$ for $x > 3$.

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State the central definition behind this outcome: Apply different rules on different input intervals and manage endpoints.

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Before calculating, list every sign, endpoint, unit, grouping, or domain condition that can affect: Evaluate $f(-2), f(1),$ and $f(4)$ when $f(x) = x + 3$ for $x < 1, f(x) = x^{2}$ for $1 \le x \le 3,$ and $f(x) = 10 - x$ for $x > 3$.

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Explain why this opening move is valid: Compare each input with the piecewise interval conditions.

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Evaluate $f(-2), f(1),$ and $f(4)$ when $f(x) = x + 3$ for $x < 1, f(x) = x^{2}$ for $1 \le x \le 3,$ and $f(x) = 10 - x$ for $x > 3$.

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Evaluate $p(-1), p(2),$ and $p(5)$ when p(x) $= 2x + 1$ for $x < 0,$ p(x) $= x^{2}$ for $0 \le x \le 3,$ and p(x) $= 10 - x$ for $x > 3$.

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Choose constants a and $b$ so q(x) $=$ ax $+ 1$ for $x < 2$ and q(x) $=$ bx $- 3$ for $x \ge 2$ has $q(2) = 5$ and matching one-sided values at $x = 2$.

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Verify the proposed result “$f(-2) = 1, f(1) = 1,$ and $f(4) = 6$.” against the original statement.

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Complete the calculation after “Match $-1$ to the first interval, $2$ to the middle interval, and $5$ to the final interval.” in this problem: Evaluate $p(-1), p(2),$ and $p(5)$ when p(x) $= 2x + 1$ for $x < 0,$ p(x) $= x^{2}$ for $0 \le x \le 3,$ and p(x) $= 10 - x$ for $x > 3$.

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Name and justify the most efficient first move, then solve: Choose constants a and $b$ so q(x) $=$ ax $+ 1$ for $x < 2$ and q(x) $=$ bx $- 3$ for $x \ge 2$ has $q(2) = 5$ and matching one-sided values at $x = 2$.

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Compare the methods used in these two cases and identify the structural reason they differ: Evaluate $p(-1), p(2),$ and $p(5)$ when p(x) $= 2x + 1$ for $x < 0,$ p(x) $= x^{2}$ for $0 \le x \le 3,$ and p(x) $= 10 - x$ for $x > 3$. Choose constants a and $b$ so q(x) $=$ ax $+ 1$ for $x < 2$ and q(x) $=$ bx $- 3$ for $x \ge 2$ has $q(2) = 5$ and matching one-sided values at $x = 2$.

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Create the representation most useful for checking this result: Evaluate $p(-1), p(2),$ and $p(5)$ when p(x) $= 2x + 1$ for $x < 0,$ p(x) $= x^{2}$ for $0 \le x \le 3,$ and p(x) $= 10 - x$ for $x > 3$. Use a mapping or table, a formula with domain, and a graph that passes the vertical-line test.

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A learner reports “$f(-2) = 1, f(1) = 1,$ and $f(4) = 6$.” but omits the original-condition check. Explain the risk before deciding whether the result is supported.

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Repair a solution that skips “Matching the first-rule approach value requires $2a + 1 = 5,$ so $a = 2$.” while solving: Choose constants a and $b$ so q(x) $=$ ax $+ 1$ for $x < 2$ and q(x) $=$ bx $- 3$ for $x \ge 2$ has $q(2) = 5$ and matching one-sided values at $x = 2$.

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In this piecewise-defined functions case, change one numerical value, solve the revised problem, and identify which parts of the original method still apply: Evaluate $f(-2), f(1),$ and $f(4)$ when $f(x) = x + 3$ for $x < 1, f(x) = x^{2}$ for $1 \le x \le 3,$ and $f(x) = 10 - x$ for $x > 3$.

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Connect the opening situation “Model postage, tax, or pricing tiers.” to the algebraic structure used in the worked case. Define quantities and units before writing any equation.

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Explain why the method for piecewise-defined functions is valid here and name one nearby problem where it would not apply.

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Compare the conclusions of all three worked cases with this lesson outcome—Apply different rules on different input intervals and manage endpoints. Explain what remains invariant across them.

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Exit check: solve and verify without referring to the displayed steps. Evaluate $p(-1), p(2),$ and $p(5)$ when p(x) $= 2x + 1$ for $x < 0,$ p(x) $= x^{2}$ for $0 \le x \le 3,$ and p(x) $= 10 - x$ for $x > 3$.

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Exit check: solve and verify without referring to the displayed steps. Choose constants a and $b$ so q(x) $=$ ax $+ 1$ for $x < 2$ and q(x) $=$ bx $- 3$ for $x \ge 2$ has $q(2) = 5$ and matching one-sided values at $x = 2$.

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Classify the mathematical object and requested action in this lesson case: Let $f(x) = \sqrt{x - 1}$ and $g(x) = \frac{1}{x - 3}$. Find the domain of $f + g$ and $\frac{f}{g}$.

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State the central definition behind this outcome: Add, subtract, multiply, and divide function outputs while tracking the combined domain.

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Before calculating, list every sign, endpoint, unit, grouping, or domain condition that can affect: Let $f(x) = \sqrt{x - 1}$ and $g(x) = \frac{1}{x - 3}$. Find the domain of $f + g$ and $\frac{f}{g}$.

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Explain why this opening move is valid: Use $x \ge 1$ for $f$ and $x \ne 3$ for $g$.

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