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Math  /  Calculus

QuestionIn Problems 15-24, find the values of x∈Rx \in \mathbf{R} for which the given functions are continuous.
15. f(x)=3x4−x2+4f(x)=3 x^{4}-x^{2}+4
16. f(x)=x2−1f(x)=\sqrt{x^{2}-1}
17. f(x)=x2+1x−1f(x)=\frac{x^{2}+1}{x-1}
18. f(x)=cos⁡(2x)f(x)=\cos (2 x)
19. f(x)=e−∣x∣f(x)=e^{-|x|}
20. f(x)=ln⁡(x−2)f(x)=\ln (x-2)
21. f(x)=ln⁡xx+1f(x)=\ln \frac{x}{x+1}
22. f(x)=exp⁡[−x−1]f(x)=\exp [-\sqrt{x-1}]

Studdy Solution

STEP 1

1. A function is continuous at a point if it is defined at that point, the limit exists at that point, and the limit equals the function value.
2. Polynomials are continuous everywhere.
3. Rational functions are continuous wherever the denominator is not zero.
4. Square root functions are continuous where the radicand is non-negative.
5. Trigonometric functions like cosine are continuous everywhere.
6. Exponential functions are continuous everywhere.
7. Logarithmic functions are continuous where the argument is positive.

STEP 2

1. Analyze the continuity of polynomial functions.
2. Analyze the continuity of square root functions.
3. Analyze the continuity of rational functions.
4. Analyze the continuity of trigonometric functions.
5. Analyze the continuity of exponential functions.
6. Analyze the continuity of logarithmic functions.

STEP 3

For problem 15, f(x)=3x4−x2+4 f(x) = 3x^4 - x^2 + 4 is a polynomial function. Polynomial functions are continuous for all x∈R x \in \mathbf{R} .

STEP 4

For problem 16, f(x)=x2−1 f(x) = \sqrt{x^2 - 1} is a square root function. The function is continuous where the radicand is non-negative:
x2−1≥0 x^2 - 1 \geq 0 x2≥1 x^2 \geq 1 x≤−1orx≥1 x \leq -1 \quad \text{or} \quad x \geq 1
The function is continuous for x∈(−∞,−1]∪[1,∞) x \in (-\infty, -1] \cup [1, \infty) .

STEP 5

For problem 17, f(x)=x2+1x−1 f(x) = \frac{x^2 + 1}{x - 1} is a rational function. It is continuous wherever the denominator is not zero:
x−1≠0 x - 1 \neq 0 x≠1 x \neq 1
The function is continuous for x∈R∖{1} x \in \mathbf{R} \setminus \{1\} .

STEP 6

For problem 18, f(x)=cos⁡(2x) f(x) = \cos(2x) is a trigonometric function. Cosine functions are continuous for all x∈R x \in \mathbf{R} .

STEP 7

For problem 19, f(x)=e−∣x∣ f(x) = e^{-|x|} is an exponential function. Exponential functions are continuous for all x∈R x \in \mathbf{R} .

STEP 8

For problem 20, f(x)=ln⁡(x−2) f(x) = \ln(x - 2) is a logarithmic function. It is continuous where the argument is positive:
x−2>0 x - 2 > 0 x>2 x > 2
The function is continuous for x∈(2,∞) x \in (2, \infty) .

STEP 9

For problem 21, f(x)=ln⁡(xx+1) f(x) = \ln\left(\frac{x}{x+1}\right) is a logarithmic function. It is continuous where the argument is positive:
xx+1>0 \frac{x}{x+1} > 0
This inequality is satisfied when both x>0 x > 0 and x+1>0 x+1 > 0 , or both x<0 x < 0 and x+1<0 x+1 < 0 . Solving these gives:
1. x>0 x > 0 and x+1>0 x+1 > 0 implies x>0 x > 0 .
2. x<0 x < 0 and x+1<0 x+1 < 0 implies x<−1 x < -1 .
The function is continuous for x∈(−∞,−1)∪(0,∞) x \in (-\infty, -1) \cup (0, \infty) .

STEP 10

For problem 22, f(x)=exp⁡[−x−1] f(x) = \exp[-\sqrt{x-1}] is an exponential function. The function is continuous where the radicand is non-negative:
x−1≥0 x - 1 \geq 0 x≥1 x \geq 1
The function is continuous for x∈[1,∞) x \in [1, \infty) .

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