Rational and Radical Functions: Get Stronger Answer Key

Rational Functions

  1. All reals [latex]x \neq 1[/latex], [latex]1[/latex]
  2. All reals [latex]x \neq -1, -2, 1, 2[/latex]
  3. V.A. at [latex]x=-\dfrac{2}{5}[/latex]; H.A. at [latex]y=0[/latex]; Domain is all reals [latex]x \neq -\dfrac{2}{5}[/latex]
  4. V.A. at [latex]x=4, -9[/latex]; H.A. at [latex]y=0[/latex]; Domain is all reals [latex]x \neq 4, -9[/latex]
  5. V.A. at [latex]x=0, 4, -4[/latex]; H.A. at [latex]y=0[/latex]; Domain is all reals [latex]x \neq 0, 4, -4[/latex]
  6. V.A. at [latex]x=5[/latex]; H.A. at [latex]y=0[/latex]; Domain is all reals [latex]x \neq 5, -5[/latex]
  7. V.A. at [latex]x=\dfrac{1}{3}[/latex]; H.A. at [latex]y=-\dfrac{2}{3}[/latex]; Domain is all reals [latex]x \neq \dfrac{1}{3}[/latex]
  8. none
  9. [latex]x[/latex]-intercepts none, [latex]y[/latex]-intercept [latex](0, \dfrac{1}{4})[/latex]
  10. Local behavior: [latex]x \to -\dfrac{1}{2}^+[/latex], [latex]f(x) \to -\infty[/latex], [latex]x \to -\dfrac{1}{2}^-[/latex], [latex]f(x) \to \infty[/latex] End behavior: [latex]x \to \pm\infty[/latex], [latex]f(x) \to \dfrac{1}{2}[/latex]
  11. Local behavior: [latex]x \to 6^+[/latex], [latex]f(x) \to -\infty[/latex], [latex]x \to 6^-[/latex], [latex]f(x) \to \infty[/latex], End behavior: [latex]x \to \pm\infty[/latex], [latex]f(x) \to -2[/latex]
  12. Local behavior: [latex]x \to \dfrac{1}{3}^+[/latex], [latex]f(x) \to -\infty[/latex], [latex]x \to \dfrac{1}{3}^-[/latex], [latex]f(x) \to \infty[/latex], [latex]x \to \dfrac{5}{2}^+[/latex], [latex]f(x) \to \infty[/latex], [latex]x \to \dfrac{5}{2}^-[/latex], [latex]f(x) \to -\infty[/latex] End behavior: [latex]x \to \pm\infty[/latex], [latex]f(x) \to \dfrac{1}{3}[/latex]
  13. [latex]y = 2x + 4[/latex]
  14. [latex]y = -2x[/latex]
  15. V.A. [latex]x=-4[/latex], H.A. [latex]y=2[/latex]; [latex](\dfrac{3}{2},0)[/latex]; [latex](0,-\dfrac{3}{4})[/latex]
    Graph of p(x)=(2x-3)/(x+4) with its vertical asymptote at x=-4 and horizontal asymptote at y=2.
  16. V.A. [latex]x=2[/latex], H.A. [latex]y=0[/latex], [latex](0,1)[/latex]
    Graph of s(x)=4/(x-2)^2 with its vertical asymptote at x=2 and horizontal asymptote at y=0.
  17. V.A. [latex]x=-4[/latex], [latex]x=\dfrac{4}{3}[/latex], H.A. [latex]y=1[/latex]; [latex](5,0)[/latex]; [latex](-\dfrac{1}{3},0)[/latex]; [latex](0,\dfrac{5}{16})[/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.
  18. V.A. [latex]x=-1[/latex], H.A. [latex]y=1[/latex]; [latex](-3,0)[/latex]; [latex](0,3)[/latex]
    Graph of a(x)=(x^2+2x-3)/(x^2-1) with its vertical asymptote at x=-1 and horizontal asymptote at y=1.
  19. [latex]y = 50\dfrac{x^2-x-2}{x^2-25}[/latex]
  20. [latex]y = 7\dfrac{x^2+2x-24}{x^2+9x+20}[/latex]
  21. [latex]y = \dfrac{1}{2}\dfrac{x^2-4x+4}{x+1}[/latex]
  22. [latex]y = 4\dfrac{x-3}{x^2-x-12}[/latex]
  23. [latex]y = \dfrac{27(x-2)}{(x+3)(x-3)^2}[/latex]
  24. [latex]C(t) = \dfrac{8+2t}{300+20t}[/latex]
  25. After about [latex]6.12[/latex] hours.
  26. [latex]A(x) = 50x^2 + \dfrac{800}{x}[/latex]. [latex]2[/latex] by [latex]2[/latex] by [latex]5[/latex] feet.
  27. [latex]A(x) = \pi x^2 + \dfrac{100}{x}[/latex]. Radius = [latex]2.52[/latex] meters.

Inverses and Radical Functions

  1. [latex]f^{-1}(x) = \sqrt{x + 4}[/latex]
  2. [latex]f^{-1}(x) = \sqrt{x + 3} - 1[/latex]
  3. [latex]f^{-1}(x) = \sqrt{12 - x}[/latex]
  4. [latex]f^{-1}(x) = \pm\sqrt{\dfrac{x-4}{2}}[/latex]
  5. [latex]f^{-1}(x) = \sqrt{\dfrac{x-1}{3}-3}[/latex]
  6. [latex]f^{-1}(x) = \sqrt{\dfrac{4-x}{2}-3}[/latex]
  7. [latex]f^{-1}(x) = \dfrac{3-x^2}{4}[/latex], [latex][0,\infty)[/latex]
  8. [latex]f^{-1}(x) = \dfrac{(x-5)^2+8}{6}[/latex]
  9. [latex]f^{-1}(x) = (3 - x)^2[/latex]
  10. [latex]f^{-1}(x) = \dfrac{4x+3}{7}[/latex]
  11. [latex]f^{-1}(x) = \dfrac{7x-3}{1-x}[/latex]
  12. [latex]f^{-1}(x) = \dfrac{2x-1}{5x+5}[/latex]
  13. [latex]f^{-1}(x) = \sqrt{x + 3} - 2[/latex]
  14. [latex]f^{-1}(x) = \sqrt{x - 2}[/latex]
    Graph showing two curves, one in orange and one in blue. The orange curve rises steeply upward in the first quadrant, starting near the origin, while the blue curve approaches the x-axis from the left and moves steadily to the right. The graph includes grid lines and labeled axes.
  15. [latex]f^{-1}(x) = \sqrt{x - 3}[/latex]
    Graph showing two intersecting curves, one in blue and one in orange. The blue curve moves upward steeply through the origin, while the orange curve has an S-shape, increasing gently as it passes through the origin and extending to the left and right. The graph includes grid lines and labeled axes.
  16. [latex]f^{-1}(x) = \sqrt[3]{x - 3}[/latex]
    Graph showing two intersecting curves, one in blue and one in orange. The blue curve rises steeply upward from left to right, passing through the origin, while the orange curve has an S-shape, gently increasing as it crosses the origin and extending horizontally to the right. The graph includes grid lines and labeled axes.
  17. [latex][-1,0] \cup [1,\infty)[/latex]
    Graph of a function f(x) with a curve that starts high on the y-axis, sharply drops near the origin, and then moves horizontally to the right as x increases. The graph includes grid lines, labeled axes, and an arrow indicating the direction of the curve.
  18. [latex][-3,0] \cup (4,\infty)[/latex]
    Graph of a function f(x) with a curve that begins high on the y-axis, drops near the origin, oscillates briefly around the x-axis, and then extends horizontally to the left. The graph includes grid lines, labeled axes, and arrows indicating the direction of the curve.
  19. [latex][-\infty,-4] \cdot [-3,3][/latex]
    Graph of an S-shaped curve passing through the origin, starting from the lower left, rising steeply, and extending to the upper right. The graph includes grid lines, labeled axes, and arrows indicating the direction of the curve.
  20. [latex]t(h) = \sqrt{\dfrac{600-h}{16}}[/latex], [latex]3.54[/latex] seconds
  21. [latex]r(A) = \sqrt{\dfrac{A}{4\pi}}[/latex], [latex]\approx 8.92[/latex] in.
  22. [latex]l(T) = 32.2\left(\dfrac{T}{2\pi}\right)[/latex], [latex]\approx 3.26[/latex]ft
  23. [latex]r(A) = \sqrt{\dfrac{A+8\pi}{2\pi}}-2[/latex], [latex]3.99[/latex] ft
  24. [latex]r(V) = \sqrt{\dfrac{V}{10\pi}}[/latex], [latex]\approx 5.64[/latex] ft

Variations

  1. [latex]y = 5x^2[/latex]
  2. [latex]y = \dfrac{1}{1944}x^3[/latex]
  3. [latex]y = 6x^4[/latex]
  4. [latex]y = \dfrac{18}{x^2}[/latex]
  5. [latex]y = \dfrac{81}{x^4}[/latex]
  6. [latex]y = \dfrac{20}{\sqrt[3]{x}}[/latex]
  7. [latex]y = 10xzw[/latex]
  8. [latex]y = 10x\sqrt{z}[/latex]
  9. [latex]y = 4\dfrac{xz}{w}[/latex]
  10. [latex]y = 40\dfrac{xz}{\sqrt{wt^2}}[/latex]
  11. [latex]y = 256[/latex]
  12. [latex]y = 6[/latex]
  13. [latex]y = 6[/latex]
  14. [latex]y = 27[/latex]
  15. [latex]y = 3[/latex]
  16. [latex]y = 18[/latex]
  17. [latex]y = 90[/latex]
  18. [latex]y = \dfrac{81}{2}[/latex]
  19. [latex]3[/latex] seconds
  20. [latex]48[/latex] inches
  21. [latex]49.75[/latex] pounds
  22. [latex]33.33[/latex] amperes
  23. [latex]2.88[/latex] inches