1.
Which statement describes the domain of the function f(x)=3x/4x^2-4?
Correct Answer
B. All real numbers except x=-1 and x=1
Explanation
The given function is a rational function. In a rational function, the denominator cannot be equal to zero. Therefore, the domain of the function is all real numbers except for the values of x that make the denominator zero. In this case, the denominator is 4x^2-4, which can be factored as 4(x^2-1). Setting this equal to zero, we get x^2-1=0, which factors as (x-1)(x+1)=0. So, the values of x that make the denominator zero are x=-1 and x=1. Therefore, the domain of the function is all real numbers except x=-1 and x=1.
2.
What is the horizontal asymptote of f(x)=-2x/x+1?
Correct Answer
A. Y= -2
Explanation
The horizontal asymptote of a function represents the value that the function approaches as x approaches positive or negative infinity. In this case, as x becomes very large or very small, the term -2x in the numerator will dominate the term x+1 in the denominator. Therefore, the function will approach -2 as x approaches infinity or negative infinity, making y = -2 the horizontal asymptote.
3.
Which statement describes the behavior of the function f(x)=2x/1-x^2?
Correct Answer
D. The graph approaches 0 as x approaches infinity.
Explanation
As x approaches infinity, the denominator x^2 becomes extremely large compared to the numerator 2x. Therefore, the fraction 2x/1-x^2 approaches 0. This can be seen by considering the limit of the function as x approaches infinity, where the numerator remains finite while the denominator grows without bound.
4.
What is the Y-intercept of y=(x+5)/(x-6)?
Correct Answer
B. (0, -5/6)
Explanation
The y-intercept of a function is the point where the graph of the function intersects the y-axis. To find the y-intercept, we substitute x=0 into the equation and solve for y. In this case, when x=0, the equation becomes y=(0+5)/(0-6) = 5/-6 = -5/6. Therefore, the y-intercept of y=(x+5)/(x-6) is (0, -5/6).
5.
What is the horizontal asymptote to this function f(x)= (4x+1)/(x+2)?
Correct Answer
A. Y=4
Explanation
The horizontal asymptote of a function represents the value that the function approaches as x approaches positive or negative infinity. In this case, as x becomes very large or very small, the term (4x+1)/(x+2) approaches 4. Therefore, the horizontal asymptote of the function f(x) = (4x+1)/(x+2) is y = 4.
6.
What is the X-Intercept of y= 4x/(x+5)?
Correct Answer
B. X=0
Explanation
The x-intercept of a function represents the point where the function intersects the x-axis. In this case, to find the x-intercept of the equation y=4x/(x+5), we set y=0 and solve for x. By substituting y=0 into the equation, we get 0=4x/(x+5). Multiplying both sides by (x+5), we eliminate the denominator and obtain 0=4x. Dividing both sides by 4, we find x=0. Therefore, the x=0 is the x-intercept of the given equation.
7.
What is an asymptote?
Correct Answer
C. An imaginary line that your function never touches
Explanation
An asymptote is an imaginary line that a function never touches. It is a line that the graph of a function approaches but does not intersect or cross. Asymptotes can be vertical, horizontal, or oblique, and they help to describe the behavior of a function as it approaches infinity or negative infinity. They provide information about the limits of a function and help to understand its overall shape and characteristics.
8.
Which asymptote(s) are determined by looking at the denominator?
Correct Answer
B. Vertical
Explanation
The asymptotes determined by looking at the denominator are the vertical asymptotes. This is because the vertical asymptotes occur at the values of x where the denominator is equal to zero. When the denominator is zero, the fraction becomes undefined, resulting in a vertical line that the graph approaches but never touches. Therefore, the correct answer is "Vertical."
9.
What is the domain of y= 1/(x+3)?
Correct Answer
A. All Reals Except -3
Explanation
The domain of a function is the set of all possible values that the independent variable (x) can take. In this case, the function is y=1/(x+3). The denominator of the fraction cannot be zero, so x+3 cannot be equal to zero. Therefore, the value of x cannot be -3. Hence, the domain of the function is all real numbers except -3.
10.
What is the Vertical Asymptotes of y= (x+5)/(x-6)?
Correct Answer
D. X=6
Explanation
The vertical asymptote of a rational function occurs at the values of x that make the denominator equal to zero. In this case, the denominator is x-6, so the vertical asymptote occurs when x-6=0. Solving for x, we find that x=6. Therefore, the vertical asymptote of the given function is x=6.