Rational Functions: Get Stronger Key — Precalculus Practice Page Answer Keys

Rational Functions Solutions

2. Yes. The numerator of the formula of the functions would have only complex roots and/or factors common to both the numerator and denominator.

3. [latex]\text{All reals }x\ne -1, 1[/latex]

4. [latex]\text{All reals }x\ne -1, -2, 1, 2[/latex]

5. V.A. at [latex]x=4, -9[/latex]; H.A. at [latex]y=0[/latex]; Domain is all reals [latex]x\ne 4, -9[/latex]

6. V.A. at [latex]x=0, 4, -4[/latex]; H.A. at [latex]y=0[/latex]; Domain is all reals [latex]x\ne 0,4, -4[/latex]

7. none

8. [latex]x\text{-intercepts none, }y\text{-intercept }\left(0,\frac{1}{4}\right)[/latex]

9. Local behavior: [latex]x\to -{\frac{1}{2}}^{+},f\left(x\right)\to -\infty ,x\to -{\frac{1}{2}}^{-},f\left(x\right)\to \infty[/latex]

End behavior: [latex]x\to \pm \infty ,f\left(x\right)\to \frac{1}{2}[/latex]

10. Local behavior: [latex]x\to {6}^{+},f\left(x\right)\to -\infty ,x\to {6}^{-},f\left(x\right)\to \infty[/latex], End behavior: [latex]x\to \pm \infty ,f\left(x\right)\to -2[/latex]

11. [latex]y=2x+4[/latex]

12. [latex]y=2x[/latex]

13. [latex]V.A.\text{ }x=-4,\text{ }H.A.\text{ }y=2;\left(\frac{3}{2},0\right);\left(0,-\frac{3}{4}\right)[/latex]
Graph of p(x)=(2x-3)/(x+4) with its vertical asymptote at x=-4 and horizontal asymptote at y=2.

14. [latex]V.A.\text{ }x=2,\text{ }H.A.\text{ }y=0,\text{ }\left(0,1\right)[/latex]
Graph of s(x)=4/(x-2)^2 with its vertical asymptote at x=2 and horizontal asymptote at y=0.

15. [latex]V.A.\text{ }x=-4,\text{ }x=\frac{4}{3},\text{ }H.A.\text{ }y=1;\left(5,0\right);\left(-\frac{1}{3},0\right);\left(0,\frac{5}{16}\right)[/latex]

16. [latex]V.A.\text{ }x=-1,\text{ }H.A.\text{ }y=1;\left(-3,0\right);\left(0,3\right)[/latex]
Graph of f(x)=(3x^2-14x-5)/(3x^2+8x-16) with its vertical asymptotes at x=-4 and x=4/3 and horizontal asymptote at y=1.

17. [latex]V.A.\text{ }x=4,\text{ }S.A.\text{ }y=2x+9;\left(-1,0\right);\left(\frac{1}{2},0\right);\left(0,\frac{1}{4}\right)[/latex]
Graph of h(x)=(2x^2+x-1)/(x-1) with its vertical asymptote at x=4 and slant asymptote at y=2x+9.

18. [latex]y=50\frac{{x}^{2}-x - 2}{{x}^{2}-25}[/latex]

19. [latex]y=7\frac{{x}^{2}+2x - 24}{{x}^{2}+9x+20}[/latex]

20. [latex]y=4\frac{x - 3}{{x}^{2}-x - 12}[/latex]

21. [latex]y=\frac{1}{3}\frac{{x}^{2}+x - 6}{x - 1}[/latex]

22. [latex]\left(\frac{3}{2},\infty \right)[/latex]
Graph of f(x)=4/(2x-3).

23. [latex]\left(-2,1\right)\cup \left(4,\infty \right)[/latex]
Graph of f(x)=(x+2)/(x-1)(x-4).

24. [latex]\left(2,4\right)[/latex]

25. [latex]\left(2,5\right)[/latex]

26. [latex]\left(-1,\text{1}\right)[/latex]

27. [latex]C\left(t\right)=\frac{8+2t}{300+20t}[/latex]

28. [latex]A\left(x\right)=50{x}^{2}+\frac{800}{x}[/latex]. 2 by 2 by 5 feet.

Modeling Using Variation Solutions

1. [latex]y=5{x}^{2}[/latex]

2. [latex]y=10{x}^{3}[/latex]

3. [latex]y=\frac{18}{{x}^{2}}[/latex]

4. [latex]y=\frac{20}{\sqrt[3]{x}}[/latex]

5. [latex]y=10xzw[/latex]

6. [latex]y=10x\sqrt{z}[/latex]

7. [latex]y=4\frac{xz}{w}[/latex]

8. [latex]y=256[/latex]

9. [latex]y=6[/latex]

10. [latex]y=18[/latex]

11. [latex]y=\frac{81}{2}[/latex]

12. 3 seconds

13. 48 inches