Question

identify the function that fits the description given below. the function that is not a continuous function. select all that apply a. y = - sin x b. y = x³ + 1 c. y = -√x d. y=(x + 2)³ e. y = - 1/x f. y = 2 - 4/(1 + e⁻ˣ) g. y = eˣ - 2 h. y=(x - 1)² i. y = ln(x + 1) j. y = -x k. y = cos x + 1 l. y = |x| - 2

Explanation:

Step1: Recall continuity rules

A function is discontinuous where it has a break, asymptote, or hole. Rational functions are discontinuous at values that make the denominator 0.

Step2: Analyze each function

• For $y =-\sin x$, $\sin x$ is continuous everywhere, so $y =-\sin x$ is continuous.
• For $y=x^{3}+1$, polynomial functions are continuous everywhere, so it is continuous.
• For $y =-\sqrt{x}$, the domain is $x\geq0$ and it is continuous on its domain.
• For $y=(x + 2)^{3}$, polynomial - continuous everywhere.
• For $y=-\frac{1}{x}$, when $x = 0$, the function is undefined. There is a vertical asymptote at $x = 0$, so it is not continuous.
• For $y=2-\frac{4}{1 + e^{-x}}$, the exponential function $e^{-x}$ is continuous everywhere and the denominator $1+e^{-x}>0$ for all real $x$, so it is continuous.
• For $y = e^{x}-2$, exponential functions are continuous everywhere, so it is continuous.
• For $y=(x - 1)^{2}$, polynomial - continuous everywhere.
• For $y=\ln(x + 1)$, the domain is $x>-1$ and it is continuous on its domain.
• For $y=-x$, linear function - continuous everywhere.
• For $y=\cos x+1$, cosine function is continuous everywhere, so it is continuous.
• For $y =|x|-2$, absolute - value function is continuous everywhere.

Answer:

E. $y =-\frac{1}{x}$