# Math Exam 3

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### D. f(x)=∛x

The graph of which of the following basic functions is increasing on the interval (-∞,∞)?

A. f(x)=1/x

B. f(x)=|x|

C. f(x)=x^2

D. f(x)=∛x

### A. y=1/x, y=x, and y∛x

Identify the collection of three functions whose graphs are all symmetric about the origin.

A. y=1/x, y=x, and y∛x

B. y=x^3, y=3, and y=1/x

C. y=|x|, y=1/x, and y=x^3

D. y=x^3, y=x^2, and y=√x

### B. y=-3, y=x^2, and y=|x|

Identify the collection of three functions that are all even.

A. y=x, y=x^2, and y=y=1/x, y=x, and y=√x

B. y=-3, y=x^2, and y=|x|

C. y=|x|, y=√x, and y=2

D. y=x^2, y=3, and y=1/x

### D. f(x)=|x|

Which of the following basic functions is equivalent to the piecewise-defined function
f(x){ x if x≥0
-x if x<0 ?

A. f(x)=1/x

B. f(x)=x^2

C. f(x)=x

D. f(x)=|x|

### A. It is possible for a piecewise-defined function to have more than one y-intercept depending on how the function is defined.

Which of the following statements is not true?

A. It is possible for a piecewise-defined function to have more than one y-intercept depending on how the function is defined.

B. Given that the graph of piecewise-defined function, it is sometimes possible to find a rule that describes the graph.

C. The range of a piecewise-defined function can be (-∞,∞).

D. The domain of a piecewise-defined function can be (-∞,∞).

### C. The graph of y=f(x)+c can be obtained by vertically shifting the graph of y=f(x) up c units.

Given the graph of y=f(x), if c is a positive real number, then which of the following statements best describes how to sketch the graph of y=f(x)+c?

A. The graph of y=f(x)+c can be obtained by horizontally shifting the graph of y=f(x) to the right c units.

B. The graph of y=f(x)+c can be obtained by vertically shifting the graph of y=f(x) down c units.

C. The graph of y=f(x)+c can be obtained by vertically shifting the graph of y=f(x) up c units.

D. The graph of y=f(x)+c can be obtained by horizontally shifting the graph of y=f(x) to the left c units.

### D. The graph of y=f(x+c) can be obtained by horizontally shifting the graph of y=f(x) to the left c units.

Given the graph of y=f(x), if c is a positive real number, then which of the following statements best describes how to sketch the graph of y=f(x+c)?

A. The graph of y=f(x+c) can be obtained by vertically shifting the graph of y=f(x) up c units.

B. The graph of y=f(x+c) can be obtained by horizontally shifting the graph of y=f(x) to the right c units.

C. The graph of y=f(x+c) can be obtained by vertically shifting the graph of y=f(x) down c units.

D. The graph of y=f(x+c) can be obtained by horizontally shifting the graph of y=f(x) to the left c units.

### C. Every function that passes the vertical line test is one-to-one.

Which of the following statements is not true?

A. Every function that passes the horizontal line test is one-to-one.

B. A function f is one-to-one if for any two range values f(u) and f(v), f(u)=f(v) implies that u=v.

C. Every function that passes the vertical line test is one-to-one.

D. A function f is one-to-one if for any values a ≠ b in the domain of f, f(a) ≠ f(b).

### B. If f has an inverse function, then f^-1(x)=1/f(x).

Which of the following statements is not true?

A. Every one-to-one function has an inverse function.

B. If f has an inverse function, then f^-1(x)=1/f(x).

C. If f and f^-1 are inverse functions, then the domain of f is the same as the range of f^-1.

D. If f and f^-1 are inverse functions, and f(a)=b, then f^-1(b)=a.

### D. If f(1)=-b, then g(b)=-1.

If f and g are inverse functions of one another, then which of the following is not necessarily true?

A. g(f(x))=x

B. If f(-a)=b, then g(b)=-a.

C. f(g(x))=x

D. If f(1)=-b, then g(b)=-1.

### A. f(x)=x^2-2x-1 and g(x)=x+1

Suppose f(g(x))=x^2. Which of the following is not a possibility for f and g?

A. f(x)=x^2-2x-1 and g(x)=x+1

B. f(x)=√x and g(x)=x^4

C. f(x)=x^2+2x+1 and g(x)=x-1

D. f(x)=x^2 and g(x)=x

### C. If x is in the domain of f and if x is in the domain of g, then x must be in the domain of f(g).

Given any two functions f and g, which of the following statements is not true?

A. The domain of f(g) is always a subset of the domain of g.

B. If x is not in the domain of g, then x cannot be in the domain of f(g).

C. If x is in the domain of f and if x is in the domain of g, then x must be in the domain of f(g).

D. To find the domain of f(g), first find the domain of g, then exclude from the domain of g all values of x for which g(x) is not in the domain of f.

### C. The constants a, b, and c must be real numbers with a not ever equal to zero.

Which of the following statements is true about the quadratic function f(x)=ax^2+bx+c?

A. The constants a, b, and c must be real numbers with a always positive.

B. The constants a, b, and c cannot ever be fractions.

C. The constants a, b, and c must be real numbers with a not ever equal to zero.

D. The constant c determines whether the graph opens up or down.

### C. The graph of f(x)=ax^2+bx+c can have either no x-intercepts or two x-intercepts but never just one x-intercept.

Which of the following statements is not true about the characteristics of the graph of f(x)=ax^2+bx+c?

A. Since the standard form of the quadratic function is f(x)=a(x-h)^2+k, the vertex always found at the point (h,k).

B. The graph of f(x)=ax^2+bx+c can never have more than one y-intercept.

C. The graph of f(x)=ax^2+bx+c can have either no x-intercepts or two x-intercepts but never just one x-intercept.

D. The axis of symmetry has the form x=h where h is the x-coordinate of the vertex.

### D. The function f(x)=-x^2+6x-8 is a possible quadratic function that describes this parabola.

Suppose that the graph of a quadratic function f is a parabola opening down and has a vertex with coordinates (3,-1). Which of the following statements is not true?

A. The equation f(x)=0 must have no real solutions.

B. The function f(x)=-1/2x^2+3x-11/2 is a possible quadratic function that describes this parabola.

C. The value f(0) must be negative.

D. The function f(x)=-x^2+6x-8 is a possible quadratic function that describes this parabola.

### C. One endpoint of the range will always be the h-coordinate of the vertex.

For a quadratic function f(x)=ax^2+bx+c, what is not true about the domain, the range, and the intercepts of the function?

A. The range will never be all real numbers.

B. The x-intercept(s) will be the real zeros of the function.

C. One endpoint of the range will always be the h-coordinate of the vertex.

D. The domain will always be all real numbers.

### D. The value of b in f(x)=ax^2+bx+c can easily be determined from the shape of the graph.

Given the graph of a quadratic function with the vertex and the y-intercept clearly identified, which of the following statements is not true?

A. The values of h and k in f(x)=a(x-h)^2+k can easily be determined because these values represent the x and y coordinates of the vertex respectively.

B. The value of c in f(x)=ax^2+bx+c can easily be determined because it represents the y-intercept of the graph.

C. The sign of the value a in f(x)=ax^2+bc+c, or equivalently f(x)=a(x-h)^2+k, can easily be determined from the shape of the graph.

D. The value of b in f(x)=ax^2+bx+c can easily be determined from the shape of the graph.

### C. The numbers an, an-1, an-2,..., a1, and a0, are positive real numbers.

Which of the following statements is not true about the polynomial function f(x)=AnX^n+A(n-1)X^(n-1)+A(n-2)X^(n-2)+...+A1X+A0?

A. The number n represents the degree of the polynomial and is a non-negative integer.

B. The number an is called a leading coefficient.

C. The numbers an, an-1, an-2,..., a1, and a0, are positive real numbers.

D. The number a0 is called the constant coefficient.

### C. If n is odd, the shape of the graph resembles a parabola.

Which of the following is not true about the shape of a power function of the form f(x)=ax^n?

A. If n=1, the graph is a straight line.

B. If a is positive and n is odd, the graph approaches negative infinity of the left side and positive infinity on the right side.

C. If n is odd, the shape of the graph resembles a parabola.

D. If a is positive and n is even, the graph approaches positive infinity on the left side and positive infinity on the right side.

### C. the sign of the leading coefficient an.

The right-hand behavior of the graph of the polynomial function of the form f(x)=AnX^n+A(n-1)X^(n-1)+A(n-2)X^(n-2)+...+A1X+A0 can be determined by

A. the number of terms in the polynomial function.

B. the sign of the constant coefficient a0.

C. the sign of the leading coefficient an.

D. the degree n of the polynomial function.

### C. examining the multiplicity of the real zeros.

The shape of the graph of a polynomial function near the x-intercepts can be determined by

A. examining whether the x-intercepts are even or odd.

B. examining the sign of the real zeros.

C. examining the multiplicity of the real zeros.

D. examining whether the x-intercepts are positive or negative.

### C. Statements II, IV, and V accurately describe this function.

For the polynomial function y=f(x) below, choose the statements that most accurately describe the function.

A. Statements I, IV, and V accurately describe this function.

B. Statements II, IV, and VI accurately describe this function.

C. Statements II, IV, and V accurately describe this function.

D. Statements II, III, and V accurately describe this function.

### B. It is possible that (x+a)<0, (x+b)<0, and (x+c)<0.

In an inequality such as (x+a)(x+b)(x+c)>0, which of the following is not a reason why setting each factor greater than zero and solving for x does not produce all of the solutions?

A. It is possible that (x+a)>0, (x+b)<0, and (x+c)<0.

B. It is possible that (x+a)<0, (x+b)<0, and (x+c)<0.

C. It is possible that (x+a)<0, (x+b)<0, and (x+c)>0.

D. It is possible that (x+a)<0, (x+b)>0, and (x+c)<0.

### B. If a polynomial inequality of the form p<0 has m distinct boundary points, then the boundary points divide the number line into m distinct intervals.

Which of the following statements is not true?

A. A boundary point of a polynomial inequality of the form p≤0 should always be represented by plotting a closed circle on a number line.

B. If a polynomial inequality of the form p<0 has m distinct boundary points, then the boundary points divide the number line into m distinct intervals.

C. A boundary point of a polynomial inequality of the form p>0 should always be represented by plotting an open circle on a number line.

D. A boundary point of a polynomial inequality of the form p<0 is a real number for which p=0.

### C. When solving a polynomial inequality, choose a test value from an interval to test whether the inequality is positive or negative on that interval.

Which of the following statements is true?

A. When solving a polynomial inequality, always determine a test value before finding any boundary points.

B. When solving a polynomial inequality, use a test value to determine if the polynomial is equal to zero.

C. When solving a polynomial inequality, choose a test value from an interval to test whether the inequality is positive or negative on that interval.

D. When solving a polynomial inequality, a test value can be the same as a boundary point.

### C. Reflect the graph of f about the line y=x to obtain the graph of f^-1.

Given the graph of a one-to-one function f, which of the following statements best describes how to sketch the graph f^-1?

A. Rotate the graph of f ninety degrees counterclockwise to obtain the graph of f^-1.

B. First reflect the graph of f about the x-axis, and then reflect the graph about the y-axis to obtain the graph of f^-1.

C. Reflect the graph of f about the line y=x to obtain the graph of f^-1.

D. Reflect the graph about the vertical line x=a for any value of a such that a=f^-1(b).

### C. If a function f has an inverse function, then we can find the inverse function by replacing f(x) with y, interchanging the variables x and y, solving for x.

Which of the following statements is not true?

A. To verify that two one-to-one functions, f and g, are inverses of each other, we must show that f(g(x))=g(f(x))=x.

B. The function f^-1 exists if and only if the function f is one-to-one.

C. If a function f has an inverse function, then we can find the inverse function by replacing f(x) with y, interchanging the variables x and y, solving for x.

D. The graph of f^-1 is a reflection of the graph of f about the line y=x.

Example: