Unit 4 Review

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

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$$g(f(3))$$

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

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Write $h$ then $g$ then $f$.

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Explain why this conclusion is valid: $t(d(100))=84.80,$ while $d(t(100))=86$. Use the foundation problem as evidence: Coupon $d(p)=p-20$ and tax $t(q)=1.06q$ at $p=100$.

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$f=2x+3,g=x^2$; $f$ after $g$.

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$g$ after $f$.

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$f=\frac{1}{x},g=x-4$; domain.

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Why parentheses?

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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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$$g(4)=1,f(1)=6$$

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$$f(-2)=5,g(5)=0$$

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Undefined versus unknown.

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First lookup in f(g(a)).

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Explain why this conclusion is valid: $f(g(2))=7$. Use the foundation problem as evidence: $g(2)=4$ and $f(4)=7$.

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Domain $sqrt(x-5)$.

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Domain $\frac{1}{x^2-16}$.

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Domain $ln(4-x^2)$.

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Two domain questions.

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Explain why this conclusion is valid: Domain $x\le -3$ or $x\ge 3$. Use the foundation problem as evidence: $f=sqrt(u),g=x^2-9$.

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Is $2x+1$ one-to-one?

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Is $x^3$ one-to-one?

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Horizontal-line test.

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Can a constant function be one-to-one on multiple inputs?

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Explain why this conclusion is valid: No; $x$ and -x share outputs. Restrict to $x\ge 0$ or $x\le 0$. Use the foundation problem as evidence: Is $x^2$ one-to-one on all real numbers?

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Inverse of $x-9$.

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Inverse of $4x$.

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Inverse of $\frac{x+3}{2}$.

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Inverse versus reciprocal.

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Explain why this conclusion is valid: $f^{-1}(x)=\frac{x+7}{3}$. Use the foundation problem as evidence: Find inverse of $3x-7$.

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Verify$$x+5 \qquad x-5$$

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

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Inverse range if original domain $[2,∞)$.

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Reflection line.

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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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Simplify$$sqrt(48)$$

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Evaluate$$16^(\frac{3}{4})$$

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State domain of $sqrt(3-x)$.

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Simplify$$sqrt(x^2)$$

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Explain why this conclusion is valid: Candidates are $3$ and $-2$; only $x=3$ satisfies the original equation. Use the foundation problem as evidence: Solve $sqrt(x+6)=x$.

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Define residual.

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Why inspect residual patterns?

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Give interpolation example.

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Give extrapolation example.

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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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