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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
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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