Which of these polynomial functions does not have a leading coefficient of 1?
k(x) = (x^2 + 4x)(x^2 - 2)
h(x) = x^3(x - 5)
f(x) = 2x^4 - x^2 + 1
✓
g(x) = (x^2 + 1)(x^2 - 3)
Correct Answer
f(x) = 2x^4 - x^2 + 1
The shared property is having a leading coefficient of 1. Options B, C, and D all expand to polynomials where the highest power of x has a coefficient of 1. Option A has a leading coefficient of 2.
Question 2
Which of these rational functions does not have a vertical asymptote at x = 3?
k(x) = (2x) / (x - 3)
h(x) = (x + 3) / (x - 3)
f(x) = (x - 1) / ((x - 3)(x + 2))
g(x) = (x^2 - 9) / (x - 3)
✓
Correct Answer
g(x) = (x^2 - 9) / (x - 3)
The shared property is having a vertical asymptote at x = 3. Options A, C, and D all have a factor of (x - 3) in the denominator that does not cancel with a factor in the numerator, leading to a vertical asymptote at x = 3. Option B simplifies to x + 3 for x ≠ 3, indicating a hole at x = 3, not a vertical asymptote.
Question 3
Which of these functions does not have a domain of all real numbers?
f(x) = 2^x
g(x) = e^(-x^2)
k(x) = 10^(x+1)
h(x) = log(x)
✓
Correct Answer
h(x) = log(x)
The shared property is having a domain of all real numbers (-∞, ∞). Options A, B, and D are exponential functions or composite functions involving exponential components, which have a domain of all real numbers. Option C is a logarithmic function, which has a domain of (0, ∞).
Question 4
Which of these logarithmic functions does not have a vertical asymptote at x = 0?
f(x) = log(x + 1)
✓
h(x) = log_2(x) - 1
k(x) = 3 log_5(x)
g(x) = ln(x)
Correct Answer
f(x) = log(x + 1)
The shared property is having a vertical asymptote at x = 0. Options B, C, and D are basic logarithmic functions or vertical transformations of them, all of which have a vertical asymptote at x = 0. Option A is a horizontal shift of log(x) to the left by 1 unit, resulting in a vertical asymptote at x = -1.
Question 5
Which of these trigonometric functions does not have a period of π?
h(x) = tan(2x)
✓
g(x) = cos(2x)
k(x) = sec(2x)
f(x) = sin(2x)
Correct Answer
h(x) = tan(2x)
The shared property is having a period of π. The period for sin(Bx), cos(Bx), csc(Bx), and sec(Bx) is 2π/|B|. For tan(Bx) and cot(Bx), it is π/|B|. Options A, B, and D all have B=2, so their period is 2π/2 = π. Option C has B=2, so its period is π/2.
Question 6
Which of these polar equations does not represent a circle?
r = 2cos(θ)
r = 4
r = 3sin(θ)
r = 1 + sin(θ)
✓
Correct Answer
r = 1 + sin(θ)
The shared property is representing a circle. Options A, B, and C all represent circles (A is centered at the origin, B and C are centered on an axis). Option D represents a cardioid.
Question 7
Which of these transformations does not result in a vertical shift of the graph of y = f(x)?
y = f(x - 2)
✓
y = f(x) - 5
y = f(x) + 3
y = f(x) + c, where c is a non-zero constant
Correct Answer
y = f(x - 2)
The shared property is resulting in a vertical shift of the graph. Options A, C, and D all involve adding or subtracting a constant to the entire function f(x), which causes a vertical shift. Option B involves subtracting a constant from the input x, which causes a horizontal shift.
Question 8
Which of these polynomial functions does not have the same end behavior as x approaches positive and negative infinity?
g(x) = x^4 - 3x^2 + 1
f(x) = x^3 - 4x
✓
k(x) = (x^2 + 1)^2
h(x) = -2x^6 + x^3
Correct Answer
f(x) = x^3 - 4x
The shared property is having the same end behavior as x approaches positive and negative infinity. This occurs when the polynomial has an even degree. Options B, C, and D all have even degrees (4, 6, and 4, respectively), so their end behaviors match. Option A has an odd degree (3), so its end behavior differs (as x→-∞, f(x)→-∞; as x→+∞, f(x)→+∞).
Question 9
Which of these rational functions does not have a horizontal asymptote at y = 0?
f(x) = 1 / (x^2 + 1)
k(x) = (x^2 + x) / (x + 1)
✓
g(x) = (2x) / (x^2 - 4)
h(x) = 3x / (x^3 + 1)
Correct Answer
k(x) = (x^2 + x) / (x + 1)
The shared property is having a horizontal asymptote at y = 0. This occurs when the degree of the denominator is greater than the degree of the numerator. Options A, B, and C all satisfy this condition. Option D has a numerator degree of 2 and a denominator degree of 1; it simplifies to x for x ≠ -1, which results in a hole at x=-1 and no horizontal asymptote.
Question 10
Which of these functions is not an inverse of an exponential function of the form y = b^x (where b > 0, b ≠ 1)?
g(x) = ln(x)
h(x) = e^x
✓
f(x) = log_2(x)
k(x) = log(x)
Correct Answer
h(x) = e^x
The shared property is being an inverse of an exponential function of the form y = b^x. Options A, B, and D are all logarithmic functions, which are the inverses of exponential functions. Option C is an exponential function itself, not an inverse of one.
Question 11
Which of these trigonometric functions is not an odd function?
g(x) = cos(x)
✓
k(x) = csc(x)
f(x) = sin(x)
h(x) = tan(x)
Correct Answer
g(x) = cos(x)
The shared property is being an odd function, meaning f(-x) = -f(x). Sine, tangent, and cosecant are all odd functions. Cosine is an even function, meaning g(-x) = g(x).
Question 12
Which of these transformations does not involve a reflection across the x-axis?
y = -f(x+1)
y = f(-x)
✓
y = -f(x)
y = -f(x) + 2
Correct Answer
y = f(-x)
The shared property is involving a reflection across the x-axis. Options B, C, and D all include a negative sign multiplying the entire function f(x) (or a transformed version of f(x)), which causes a reflection across the x-axis. Option A involves a negative sign multiplying only the input x, which causes a reflection across the y-axis.
Question 13
Which of these polynomial functions does not have exactly one real root?
h(x) = x^5 - 2
f(x) = x^3 + 1
g(x) = x^4 + 1
✓
k(x) = x^3 - 8
Correct Answer
g(x) = x^4 + 1
The shared property is having exactly one real root. Options A, C, and D are odd-degree polynomials that can be solved to find exactly one real root (x=-1, x=2^(1/5), and x=2 respectively). Option B is an even-degree polynomial that is always positive (since x^4 ≥ 0, then x^4 + 1 ≥ 1), thus having no real roots.
Question 14
Which of these polar equations does not exhibit symmetry with respect to the polar axis (x-axis)?
r = 5
r = 3cos(θ)
r = 1 + sin(θ)
✓
r = 2 + cos(θ)
Correct Answer
r = 1 + sin(θ)
The shared property is exhibiting symmetry with respect to the polar axis. A polar graph has symmetry with respect to the polar axis if replacing θ with -θ results in an equivalent equation. For options A, B, and D (since cos(-θ) = cos(θ) and r=5 is a circle centered at the origin), substituting -θ for θ yields an equivalent equation. For option C, r = 1 + sin(-θ) = 1 - sin(θ), which is not equivalent to r = 1 + sin(θ), indicating no symmetry with respect to the polar axis.
Question 15
Which of these rational functions does not have an oblique (slant) asymptote?
g(x) = (x^3 - 2x) / (x^2 + 1)
f(x) = (x^2 + 1) / (x - 2)
k(x) = (x^2 + 4) / (x^2 - 1)
✓
h(x) = (2x^2 + 3x - 1) / (x + 1)
Correct Answer
k(x) = (x^2 + 4) / (x^2 - 1)
The shared property is having an oblique (slant) asymptote. An oblique asymptote exists when the degree of the numerator is exactly one greater than the degree of the denominator. Options A, B, and C all satisfy this condition. Option D has the degree of the numerator equal to the degree of the denominator, which results in a horizontal asymptote (y=1), not an oblique asymptote.