Questions: Find the domain and the x- and y-intercepts of f(x). Then graph the function. f(x) = (x^3 + 5x^2 - 14x) / (x^2 - 2x - 24)

Find the domain and the x- and y-intercepts of f(x). Then graph the function.
f(x) = (x^3 + 5x^2 - 14x) / (x^2 - 2x - 24)
Transcript text: Find the domain and the $x$ - and $y$-intercepts of $f(x)$. Then graph the function. \[ f(x)=\frac{x^{3}+5 x^{2}-14 x}{x^{2}-2 x-24} \]
failed

Solution

failed
failed

Solution Steps

Step 1: Find the Domain of f(x) f(x)

The function f(x)=x3+5x214xx22x24 f(x) = \frac{x^{3}+5x^{2}-14x}{x^{2}-2x-24} is undefined where the denominator is zero. To find these points, solve the equation:

x22x24=0 x^{2} - 2x - 24 = 0

Using the quadratic formula x=b±b24ac2a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} , where a=1 a = 1 , b=2 b = -2 , and c=24 c = -24 :

x=(2)±(2)241(24)21 x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-24)}}{2 \cdot 1} x=2±4+962 x = \frac{2 \pm \sqrt{4 + 96}}{2} x=2±1002 x = \frac{2 \pm \sqrt{100}}{2} x=2±102 x = \frac{2 \pm 10}{2}

Thus, x=6 x = 6 or x=4 x = -4 .

The domain of f(x) f(x) is all real numbers except x=6 x = 6 and x=4 x = -4 .

Step 2: Find the x x -intercepts

The x x -intercepts occur where f(x)=0 f(x) = 0 , which is when the numerator is zero:

x3+5x214x=0 x^{3} + 5x^{2} - 14x = 0

Factor out an x x :

x(x2+5x14)=0 x(x^{2} + 5x - 14) = 0

The solutions are x=0 x = 0 or solving x2+5x14=0 x^{2} + 5x - 14 = 0 using the quadratic formula:

x=5±5241(14)21 x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-14)}}{2 \cdot 1} x=5±25+562 x = \frac{-5 \pm \sqrt{25 + 56}}{2} x=5±812 x = \frac{-5 \pm \sqrt{81}}{2} x=5±92 x = \frac{-5 \pm 9}{2}

Thus, x=2 x = 2 or x=7 x = -7 .

The x x -intercepts are x=0 x = 0 , x=2 x = 2 , and x=7 x = -7 .

Step 3: Find the y y -intercept

The y y -intercept occurs where x=0 x = 0 :

f(0)=03+502140022024=0 f(0) = \frac{0^{3} + 5 \cdot 0^{2} - 14 \cdot 0}{0^{2} - 2 \cdot 0 - 24} = 0

The y y -intercept is y=0 y = 0 .

Final Answer

  • Domain: xR{6,4} x \in \mathbb{R} \setminus \{6, -4\}
  • x x -intercepts: x=0 x = 0 , x=2 x = 2 , x=7 x = -7
  • y y -intercept: y=0 y = 0

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -10, "ymax": 10}, "commands": ["y = (x3 + 5*x2 - 14_x)/(x**2 - 2_x - 24)"], "latex_expressions": ["y=fracx3+5x214xx22x24y = \\frac{x^{3}+5x^{2}-14x}{x^{2}-2x-24}"]}

Was this solution helpful?
failed
Unhelpful
failed
Helpful