Unit 4 Flexible Practice

$f=x+10,g=2x$; $f(g(3))$.

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Which acts first in f(g(x))?

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Solve the representation example, then name the feature of composition as sequential processing that it illustrates$$g(f(3))$$

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Create a nearby example by changing one number or condition in this prompt: Which acts first in f(g(x))? Predict the effect, solve your new example, and compare it with the original.

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$g$ after $f$.

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Explain why this conclusion is valid: $f(g(x))=9x^2+12x+3$. Use the foundation problem as evidence: $f=x^2-1,g=3x+2$.

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Correct this reasoning and identify the first unsafe assumption: A frequent error is substituting into only one term or dropping grouping symbols.

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Write a short verification checklist for composition from formulas, then apply it to one worked example from this lesson.

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Undefined versus unknown.

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Solve the representation example, then name the feature of composition from tables and graphs that it illustrates$$f(-2)=5,g(5)=0$$

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Create a nearby example by changing one number or condition in this prompt: Undefined versus unknown. Predict the effect, solve your new example, and compare it with the original.

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Domain $sqrt(x-5)$.

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Two domain questions.

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Correct this reasoning and identify the first unsafe assumption: A frequent error is simply intersecting the written domains of $f$ and $g$ as though they used the same stage variable.

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Write a short verification checklist for domains of composite functions, then apply it to one worked example from this lesson.

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Horizontal-line test.

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Solve the representation example, then name the feature of one-to-one functions that it illustrates: Is $x^3$ one-to-one?

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Connect two representations for this example: Is $x^2$ one-to-one on all real numbers? Describe what a graph, table, mapping, or algebraic form would have to show.

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Inverse of $x-9$.

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Inverse versus reciprocal.

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Correct this reasoning and identify the first unsafe assumption: A frequent error is keeping both plus and minus branches and producing an inverse relation rather than a function.

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Write a short verification checklist for constructing inverse functions, then apply it to one worked example from this lesson.

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Reflect $(3,-1)$ across $y=x$.

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Explain why this conclusion is valid: Both compositions simplify to $x$. Use the foundation problem as evidence: Verify $f=3x-2$ and $g=\frac{x+2}{3}$.

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Connect two representations for this example: Verify $f=3x-2$ and $g=\frac{x+2}{3}$. Describe what a graph, table, mapping, or algebraic form would have to show.

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Simplify$$sqrt(48)$$

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Simplify$$sqrt(x^2)$$

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Solve the representation example, then name the feature of domain restrictions and radical inverses that it illustrates: Evaluate$$16^(\frac{3}{4})$$

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Create a nearby example by changing one number or condition in this prompt: State domain of $sqrt(3-x)$. Predict the effect, solve your new example, and compare it with the original.

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Why inspect residual patterns?

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Explain why this conclusion is valid: $C(h)=40+25h$. Use the foundation problem as evidence: Costs $65,90,115,140$ for $1-4$ hours.

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Connect two representations for this example: Costs $65,90,115,140$ for $1-4$ hours. Describe what a graph, table, mapping, or algebraic form would have to show.

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