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Math  /  Algebra

QuestionIn Exercises 1-4, find the domain of the function ff. Use limits to describe the behavior of ff at value(s) of xx not in its domain.
1. f(x)=1x+3f(x)=\frac{1}{x+3}
2. f(x)=−3x−1f(x)=\frac{-3}{x-1}
3. f(x)=−1x2−4f(x)=\frac{-1}{x^{2}-4}
4. f(x)=2x2−1f(x)=\frac{2}{x^{2}-1}

Studdy Solution

STEP 1

1. The domain of a function is the set of all possible input values (x-values) for which the function is defined.
2. A rational function is undefined where its denominator is zero.
3. We will use limits to describe the behavior of the function near points where it is undefined.

STEP 2

1. Determine the domain of f(x)=1x+3 f(x) = \frac{1}{x+3} .
2. Use limits to describe the behavior of f(x)=1x+3 f(x) = \frac{1}{x+3} at points not in its domain.
3. Determine the domain of f(x)=−3x−1 f(x) = \frac{-3}{x-1} .
4. Use limits to describe the behavior of f(x)=−3x−1 f(x) = \frac{-3}{x-1} at points not in its domain.
5. Determine the domain of f(x)=−1x2−4 f(x) = \frac{-1}{x^2-4} .
6. Use limits to describe the behavior of f(x)=−1x2−4 f(x) = \frac{-1}{x^2-4} at points not in its domain.
7. Determine the domain of f(x)=2x2−1 f(x) = \frac{2}{x^2-1} .
8. Use limits to describe the behavior of f(x)=2x2−1 f(x) = \frac{2}{x^2-1} at points not in its domain.

STEP 3

Determine the domain of f(x)=1x+3 f(x) = \frac{1}{x+3} :
The function is undefined where the denominator is zero. Solve for x x :
x+3=0 x + 3 = 0 x=−3 x = -3
Thus, the domain is all real numbers except x=−3 x = -3 .

STEP 4

Use limits to describe the behavior of f(x)=1x+3 f(x) = \frac{1}{x+3} at x=−3 x = -3 :
Calculate the limits as x x approaches −3-3 from the left and right:
lim⁡x→−3−1x+3=−∞ \lim_{x \to -3^-} \frac{1}{x+3} = -\infty lim⁡x→−3+1x+3=+∞ \lim_{x \to -3^+} \frac{1}{x+3} = +\infty
The function approaches negative infinity from the left and positive infinity from the right at x=−3 x = -3 .

STEP 5

Determine the domain of f(x)=−3x−1 f(x) = \frac{-3}{x-1} :
The function is undefined where the denominator is zero. Solve for x x :
x−1=0 x - 1 = 0 x=1 x = 1
Thus, the domain is all real numbers except x=1 x = 1 .

STEP 6

Use limits to describe the behavior of f(x)=−3x−1 f(x) = \frac{-3}{x-1} at x=1 x = 1 :
Calculate the limits as x x approaches 11 from the left and right:
lim⁡x→1−−3x−1=+∞ \lim_{x \to 1^-} \frac{-3}{x-1} = +\infty lim⁡x→1+−3x−1=−∞ \lim_{x \to 1^+} \frac{-3}{x-1} = -\infty
The function approaches positive infinity from the left and negative infinity from the right at x=1 x = 1 .

STEP 7

Determine the domain of f(x)=−1x2−4 f(x) = \frac{-1}{x^2-4} :
The function is undefined where the denominator is zero. Solve for x x :
x2−4=0 x^2 - 4 = 0 (x−2)(x+2)=0 (x-2)(x+2) = 0 x=2orx=−2 x = 2 \quad \text{or} \quad x = -2
Thus, the domain is all real numbers except x=2 x = 2 and x=−2 x = -2 .

STEP 8

Use limits to describe the behavior of f(x)=−1x2−4 f(x) = \frac{-1}{x^2-4} at x=2 x = 2 and x=−2 x = -2 :
Calculate the limits as x x approaches 22 from the left and right:
lim⁡x→2−−1x2−4=−∞ \lim_{x \to 2^-} \frac{-1}{x^2-4} = -\infty lim⁡x→2+−1x2−4=−∞ \lim_{x \to 2^+} \frac{-1}{x^2-4} = -\infty
Calculate the limits as x x approaches −2-2 from the left and right:
lim⁡x→−2−−1x2−4=−∞ \lim_{x \to -2^-} \frac{-1}{x^2-4} = -\infty lim⁡x→−2+−1x2−4=−∞ \lim_{x \to -2^+} \frac{-1}{x^2-4} = -\infty
The function approaches negative infinity from both sides at x=2 x = 2 and x=−2 x = -2 .

STEP 9

Determine the domain of f(x)=2x2−1 f(x) = \frac{2}{x^2-1} :
The function is undefined where the denominator is zero. Solve for x x :
x2−1=0 x^2 - 1 = 0 (x−1)(x+1)=0 (x-1)(x+1) = 0 x=1orx=−1 x = 1 \quad \text{or} \quad x = -1
Thus, the domain is all real numbers except x=1 x = 1 and x=−1 x = -1 .

STEP 10

Use limits to describe the behavior of f(x)=2x2−1 f(x) = \frac{2}{x^2-1} at x=1 x = 1 and x=−1 x = -1 :
Calculate the limits as x x approaches 11 from the left and right:
lim⁡x→1−2x2−1=+∞ \lim_{x \to 1^-} \frac{2}{x^2-1} = +\infty lim⁡x→1+2x2−1=−∞ \lim_{x \to 1^+} \frac{2}{x^2-1} = -\infty
Calculate the limits as x x approaches −1-1 from the left and right:
lim⁡x→−1−2x2−1=−∞ \lim_{x \to -1^-} \frac{2}{x^2-1} = -\infty lim⁡x→−1+2x2−1=+∞ \lim_{x \to -1^+} \frac{2}{x^2-1} = +\infty
The function approaches positive infinity from the left and negative infinity from the right at x=1 x = 1 , and negative infinity from the left and positive infinity from the right at x=−1 x = -1 .
The domains and behaviors are as follows:
1. Domain: All real numbers except x=−3 x = -3 . Behavior: Approaches −∞-\infty from the left and +∞+\infty from the right at x=−3 x = -3 .
2. Domain: All real numbers except x=1 x = 1 . Behavior: Approaches +∞+\infty from the left and −∞-\infty from the right at x=1 x = 1 .
3. Domain: All real numbers except x=2 x = 2 and x=−2 x = -2 . Behavior: Approaches −∞-\infty from both sides at x=2 x = 2 and x=−2 x = -2 .
4. Domain: All real numbers except x=1 x = 1 and x=−1 x = -1 . Behavior: Approaches +∞+\infty from the left and −∞-\infty from the right at x=1 x = 1 , and −∞-\infty from the left and +∞+\infty from the right at x=−1 x = -1 .

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