Q. Which of the following best describes a 'piecewise function' as mentioned in the passage?
A.
A function defined by different expressions for different intervals.
B.
A function that is continuous everywhere.
C.
A function that has only one expression.
D.
A function that is defined only at discrete points.
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Solution
A piecewise function is defined by different expressions depending on the interval of the input value.
Correct Answer:
A
— A function defined by different expressions for different intervals.
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Q. Which of the following best describes a function that is one-to-one?
A.
Each output is paired with exactly one input.
B.
Each input corresponds to multiple outputs.
C.
The graph is symmetric about the origin.
D.
The function is always increasing.
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Solution
A one-to-one function has each output paired with exactly one input, meaning it passes the horizontal line test.
Correct Answer:
A
— Each output is paired with exactly one input.
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Q. Which of the following best describes a function that is periodic?
A.
It has a constant value.
B.
It repeats its values at regular intervals.
C.
It is always increasing.
D.
It has no maximum or minimum values.
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Solution
A periodic function is one that repeats its values at regular intervals, such as sine and cosine functions.
Correct Answer:
B
— It repeats its values at regular intervals.
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Q. Which of the following best describes the 'slope' of a linear function as mentioned in the passage?
A.
The rate of change of the function.
B.
The maximum value of the function.
C.
The y-intercept of the function.
D.
The area under the curve.
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Solution
The slope of a linear function indicates the rate of change of the function with respect to x.
Correct Answer:
A
— The rate of change of the function.
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Q. Which of the following best describes the end behavior of the function f(x) = -x^4?
A.
Both ends go to positive infinity.
B.
Both ends go to negative infinity.
C.
The left end goes to negative infinity and the right end goes to positive infinity.
D.
The left end goes to positive infinity and the right end goes to negative infinity.
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Solution
Since the leading coefficient is negative and the degree is even, both ends of the graph will go to negative infinity.
Correct Answer:
B
— Both ends go to negative infinity.
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Q. Which of the following describes the end behavior of the graph of a cubic function?
A.
Both ends rise.
B.
Both ends fall.
C.
One end rises and the other falls.
D.
The graph is constant.
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Solution
A cubic function has one end rising and the other falling, characteristic of odd-degree polynomials.
Correct Answer:
C
— One end rises and the other falls.
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Q. Which of the following describes the range of the function f(x) = x^2?
A.
All real numbers.
B.
All positive real numbers.
C.
All non-negative real numbers.
D.
All integers.
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Solution
The range of f(x) = x^2 is all non-negative real numbers since the output of the function is always zero or positive.
Correct Answer:
C
— All non-negative real numbers.
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Q. Which of the following functions has a graph that approaches but never touches the x-axis?
A.
Linear function
B.
Quadratic function
C.
Exponential function
D.
Constant function
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Solution
An exponential function approaches the x-axis as x approaches negative infinity but never actually touches it.
Correct Answer:
C
— Exponential function
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Q. Which of the following functions has a vertical asymptote?
A.
f(x) = x^2 + 1
B.
f(x) = 1/(x - 2)
C.
f(x) = e^x
D.
f(x) = log(x)
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Solution
The function f(x) = 1/(x - 2) has a vertical asymptote at x = 2, where the function is undefined.
Correct Answer:
B
— f(x) = 1/(x - 2)
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Q. Which of the following graphs represents a function that is neither increasing nor decreasing?
A.
A straight line with a positive slope
B.
A straight line with a negative slope
C.
A horizontal line
D.
A parabolic curve opening upwards
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Solution
A horizontal line represents a function that is constant, meaning it neither increases nor decreases.
Correct Answer:
C
— A horizontal line
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Q. Which of the following graphs represents a function that is not one-to-one?
A.
A straight line with a positive slope.
B.
A parabola opening upwards.
C.
A horizontal line.
D.
A line with a negative slope.
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Solution
A parabola opening upwards is not one-to-one because it fails the horizontal line test; a horizontal line can intersect it at two points.
Correct Answer:
B
— A parabola opening upwards.
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Q. Which of the following graphs represents a function that is strictly increasing?
A.
A graph that slopes downward from left to right.
B.
A graph that slopes upward from left to right.
C.
A horizontal line.
D.
A graph that has both increasing and decreasing intervals.
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Solution
A strictly increasing function is one where, for any two points x1 and x2, if x1 < x2, then f(x1) < f(x2), which is represented by a graph that slopes upward from left to right.
Correct Answer:
B
— A graph that slopes upward from left to right.
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Q. Which of the following graphs represents a quadratic function?
A.
A straight line.
B.
A parabola opening upwards or downwards.
C.
A hyperbola.
D.
A circle.
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Solution
A quadratic function is represented by a parabola, which can open either upwards or downwards.
Correct Answer:
B
— A parabola opening upwards or downwards.
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Q. Which of the following is a characteristic of exponential functions?
A.
They have a constant rate of change.
B.
They grow or decay at a constant percentage rate.
C.
They are always positive.
D.
They can be represented by a straight line.
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Solution
Exponential functions grow or decay at a constant percentage rate, which is a defining characteristic.
Correct Answer:
B
— They grow or decay at a constant percentage rate.
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Q. Which of the following is true about the roots of a cubic function?
A.
It can have at most two real roots.
B.
It can have at most three real roots.
C.
It can have no real roots.
D.
It must have at least one real root.
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Solution
A cubic function can have at most three real roots, and it is guaranteed to have at least one real root due to the Intermediate Value Theorem.
Correct Answer:
B
— It can have at most three real roots.
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Q. Which of the following statements about the graph of a function is true if it is continuous everywhere?
A.
It has no breaks or holes.
B.
It must be a polynomial function.
C.
It can only be a linear function.
D.
It must have at least one x-intercept.
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Solution
A continuous function has no breaks or holes in its graph.
Correct Answer:
A
— It has no breaks or holes.
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Q. Which of the following statements about the inverse of a function is true?
A.
The inverse of a function is always a function.
B.
The inverse of a function is symmetric to the original function about the line y = x.
C.
The inverse can only exist for polynomial functions.
D.
The inverse of a function is always linear.
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Solution
The inverse of a function is symmetric to the original function about the line y = x, provided the original function is one-to-one.
Correct Answer:
B
— The inverse of a function is symmetric to the original function about the line y = x.
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Q. Which of the following statements is true about the graph of a function that is periodic?
A.
It repeats its values at regular intervals.
B.
It is always increasing.
C.
It has no maximum or minimum values.
D.
It is a straight line.
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Solution
A periodic function is characterized by repeating values at regular intervals, such as sine and cosine functions.
Correct Answer:
A
— It repeats its values at regular intervals.
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Q. Which of the following statements is true about the inverse of a function?
A.
The inverse of a function is always a function.
B.
The inverse of a function is not necessarily a function.
C.
The inverse of a function is always linear.
D.
The inverse of a function cannot be graphed.
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Solution
The inverse of a function is a function only if the original function is one-to-one.
Correct Answer:
B
— The inverse of a function is not necessarily a function.
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Q. Which of the following statements is true about the roots of a polynomial function?
A.
A polynomial can have at most as many roots as its degree.
B.
All roots of a polynomial are real numbers.
C.
A polynomial of degree n has exactly n distinct roots.
D.
Roots of a polynomial cannot be complex.
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Solution
A polynomial function can have at most as many roots as its degree, but not all roots need to be real.
Correct Answer:
A
— A polynomial can have at most as many roots as its degree.
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Q. Which of the following statements is true regarding the composition of functions?
A.
The composition of two functions is always a function.
B.
The composition of two functions is never a function.
C.
The composition of two functions can be a function or not, depending on the functions involved.
D.
The composition of two functions is always linear.
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Solution
The composition of two functions can result in a function or not, depending on the nature of the original functions.
Correct Answer:
C
— The composition of two functions can be a function or not, depending on the functions involved.
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Q. Which of the following statements is true regarding the inverse of a function?
A.
The inverse of a function is always a function.
B.
The inverse of a function is defined only for linear functions.
C.
The graph of a function and its inverse are symmetric about the line y = x.
D.
The inverse of a function can never be found.
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Solution
The graph of a function and its inverse are symmetric about the line y = x, which is a key property of inverse functions.
Correct Answer:
C
— The graph of a function and its inverse are symmetric about the line y = x.
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Q. Which of the following statements is true regarding the vertical line test for functions?
A.
A vertical line can intersect a graph at more than one point for it to be a function.
B.
A vertical line can intersect a graph at only one point for it to be a function.
C.
A vertical line test is irrelevant for determining functions.
D.
All graphs pass the vertical line test.
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Solution
For a graph to represent a function, any vertical line drawn must intersect the graph at most once.
Correct Answer:
B
— A vertical line can intersect a graph at only one point for it to be a function.
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Q. Which of the following transformations would result in a vertical shift of the graph of the function f(x) = x^2?
A.
f(x) = x^2 + 3
B.
f(x) = 3x^2
C.
f(x) = x^2 - 3
D.
f(x) = 2x^2
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Solution
Adding a constant to the function, such as f(x) = x^2 + 3, results in a vertical shift upwards.
Correct Answer:
A
— f(x) = x^2 + 3
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Q. Which of the following transformations would result in a vertical shift of the graph of the function f(x)?
A.
f(x) + k
B.
f(x + k)
C.
kf(x)
D.
f(kx)
Show solution
Solution
Adding a constant k to the function f(x) results in a vertical shift of the graph.
Correct Answer:
A
— f(x) + k
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