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Lesson 5: Solving Rational Equations — Practice Questions

  1. 1. Find the inverse function, $f^{-1}(x)$, for the rational function $f(x) = \frac{4x - 1}{x + 5}$. $f^{-1}(x) = $ ___

  2. 2. To find the inverse of $g(x) = \frac{3x + 7}{2x - 5}$, what is the correct equation after replacing $g(x)$ with $y$ and switching the variables?

    • A. $x = \frac{3y + 7}{2y - 5}$
    • B. $y = \frac{3x + 7}{2x - 5}$
    • C. $x = \frac{2y - 5}{3y + 7}$
    • D. $x = \frac{3x + 7}{2y - 5}$
  3. 3. Given the function $h(x) = \frac{3x + 2}{x - 3}$, what is its inverse, $h^{-1}(x)$? $h^{-1}(x) = $ ___

  4. 4. Which of the following functions is the inverse of $f(x) = \frac{5x - 3}{x + 2}$?

    • A. $g(x) = \frac{-2x - 3}{x - 5}$
    • B. $g(x) = \frac{x + 2}{5x - 3}$
    • C. $g(x) = \frac{2x + 3}{x - 5}$
    • D. $g(x) = \frac{5x + 3}{x - 2}$
  5. 5. When finding the inverse of $f(x) = \frac{x - 7}{3x + 2}$, we can isolate terms with $y$ to get the equation $y(3x - 1) = A$. What is the expression for $A$? $A = $ ___

  6. 6. Solve for $r$: $\frac{12}{r - 7} = 3$. \newline $r = $ ___.

  7. 7. Solve for $m$: $\frac{3}{m^2} = 5$.

    • A. $m = \pm \frac{\sqrt{15}}{5}$
    • B. $m = \pm \sqrt{\frac{5}{3}}$
    • C. $m = \pm \frac{3}{5}$
    • D. $m = \frac{\sqrt{15}}{5}$
  8. 8. Solve for $x$: $\frac{15}{x^2} = 8$.

    • A. $x = \pm \frac{\sqrt{30}}{4}$
    • B. $x = \pm \frac{\sqrt{15}}{8}$
    • C. $x = \pm \frac{15}{8}$
    • D. $x = \frac{\sqrt{30}}{4}$
  9. 9. Solve for $v$: $-3 = \frac{v + 1}{v - 6}$. \newline $v = $ ___.

  10. 10. Solve for $y$: $4.3 = \sqrt{\frac{18}{y}}$. Round your answer to one decimal place. \newline $y = $ ___.