Section 1.5: Rational Expressions

  1. Find the domain of the following functions in interval notation.   solution

    2. (a)   \( \displaystyle{f(x)=\frac{3}{x+2}} \)  (b)   \(\displaystyle{f(x)=\frac{x^2+9}{x^2-9}}\)  (c)  \(\displaystyle{f(x)=x^2-x+1}\)   (d) \( \displaystyle{ f(x)= \frac{2}{\sqrt{4-x^2}} }\) (e)   \(\displaystyle{ f(x)= \frac{\sqrt{4-2x}}{x+5}}\)

  2. Find the domain of the function.   solution

    3. (a) \(g(x)=\sqrt{x+5} \)   (b) \(f(x)=\dfrac{x-5}{\sqrt{43+8x}} \)

  3. Simplify the rational expression.  solution

    4. (a) \(\dfrac{x^{2}+3 x+2}{x^{2}-x-2}\)   (b) \(\dfrac{2 x^{2}-x-1}{x^{2}-9} \cdot \dfrac{x+3}{2 x+1}\)  (c) \(\dfrac{\frac{y}{x}-\frac{x}{y}}{\frac{1}{y}-\frac{1}{x}}\)

  4. Simplify.  solution

    5. (a) \(\dfrac{x^3-9x}{2x-6} \)   (b) \(\dfrac{\frac{1}{2+h}-\frac{1}{2}}{h} \)   (c) \(\displaystyle{ \frac{(5+h)^2-25}{h}} \)

  5. Simplify.  solution

    6. (a) \(\dfrac{1}{x}+\dfrac{1}{x-2} \)   (b) \( \dfrac{1}{x+2}-\dfrac{x}{(x+2)(x-2)} \)   (c) \( \dfrac{1}{x^2+x}-\dfrac{1}{x^2-1} \)

  6. Perform the multiplication or division and simplify.   solution

    7. (a) \(\dfrac{x^2+3 x-40}{x^2-3 x-40} \cdot \dfrac{8-x}{8+x} \)   (b) \(\dfrac{\frac{2 x^2-9 x-5}{x^2-16}}{\frac{2 x^2+11 x+5}{x^2+x-20}} \)

  7. Simplify the fractional expression.   solution

    8. (a) \( \dfrac{\frac{1}{6+x+h}-\frac{1}{6+x}}{h} \)   (b) \( \dfrac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h} \)

  8. Rationalize the numerator. (Simplify your answer completely.)   solution

    9. (a) \( \dfrac{\sqrt{25+h}-5}{h}\)    (b) \( \dfrac{\sqrt{9-h}-3}{h}\)

  9. Rationalize the numerator. (Simplify your answer completely.)   solution

    10. (a) \( \dfrac{\sqrt{3}+\sqrt{5}}{4}\)   (b) \( \dfrac{\sqrt{x}-\sqrt{x+h}}{h \sqrt{x} \sqrt{x+h}}\)

  10. Simplify the compound fractional expression.   solution

    11. (a) \(\dfrac{1+\frac{1}{x}}{\frac{1}{x}-5}\)   (b) \(\dfrac{1+\frac{1}{x+7}}{1-\frac{1}{x+7}}\)