Unit 6 Review

Domain of $\frac{1}{x+5}$.

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Domain of $\frac{x-2}{x^2-9}$.

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Do numerator zeros restrict domain?

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Can rational function have all-real domain?

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Explain why this conclusion is valid: $x\ne 3,-2$. Use the foundation problem as evidence: Domain of $\frac{x}{x^2-x-6}$.

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Hole of $\frac{x^2-1}{x-1}$.

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Hole of $\frac{x^2+3x+2}{x+1}$.

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Can a function have two holes?

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What sets hole y-value?

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Explain why this conclusion is valid: Line $y=x+3$ with hole $(3,6)$. Use the foundation problem as evidence: Analyze $\frac{x^2-9}{x-3}$.

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VAs of $\frac{1}{x^2-9}$.

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Does fully cancelled zero create VA?

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Can graph include asymptote x-value?

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Explain why this conclusion is valid: Left -∞, right $+∞$. Use the foundation problem as evidence: Analyze $\frac{1}{x-2}$ near $2$.

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HA of $\frac{1}{x+2}$.

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HA of $\frac{3x^2+1}{x^2-4}$.

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Does $\frac{x^3+1}{x^2+1}$ have HA?

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How test whether HA level occurs?

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Explain why this conclusion is valid: Horizontal asymptote $y=\frac{5}{2}$. Use the foundation problem as evidence: Analyze $\frac{5x^3-2}{2x^3+x}$.

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Slant of $\frac{x^2+x+1}{x}$.

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When slant occurs.

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Can graph cross it?

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Explain why this conclusion is valid: $x+1+\frac{2}{x-1},$ so asymptote $y=x+1$. Use the foundation problem as evidence: Analyze $\frac{x^2+1}{x-1}$.

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X-intercept of $\frac{x-5}{x+1}$.

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Y-intercept.

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Critical values of $\frac{x}{(x-2)(x+4)}$.

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Can a hole be intercept?

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Explain why this conclusion is valid: Positive, negative, positive across $-2$ and $1$. Use the foundation problem as evidence: Analyze signs of $\frac{x-1}{x+2}$.

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Features of $\frac{1}{x-4}$.

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Can HA be crossed?

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Why selected points?

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Explain why this conclusion is valid: VA $x=2,$ HA $y=1,$ x-intercept $-1,$ y-intercept $-\frac{1}{2,}$ signs $+,-,+$. Use the foundation problem as evidence: Analyze $\frac{x+1}{x-2}$.

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Solve$$\frac{1}{x}=\frac{1}{4}$$

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Solve$$\frac{2}{x}=\frac{1}{x-3}$$

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X-intercept of $\frac{x-2}{x+3}$.

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How find intersection $y$?

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Explain why this conclusion is valid: Candidate $x=2$ is excluded, so no solution. Use the foundation problem as evidence: Solve $\frac{x+1}{x-2}=\frac{3}{x-2}$.

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Solve$$\frac{x-2}{x+5}<0$$

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Solve$$\frac{1}{x}<0$$

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Which endpoints may be included?

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Explain why this conclusion is valid: $[-3,4)$. Use the foundation problem as evidence: Solve $\frac{x+3}{x-4}\le 0$.

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Direct $y=14$ at $x=2$.

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Inverse $y=3$ at $x=10$.

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If $x$ doubles in $\frac{k}{x}$.

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Why $x=0$ excluded?

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Explain why this conclusion is valid: $y=\frac{40}{x}$. Use the foundation problem as evidence: $y$ varies inversely with $x$; $y=5$ at $x=8$.

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Form for $\frac{1}{(x-1)(x+3)}$.

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Is $\frac{x^2+1}{x-1}$ proper?

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What first if improper?

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Can decomposition restore exclusions?

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