Algebra and Functions: Master Advanced Algebraic Concepts

Question 1

To translate the graph of f(x) = x^2 three units to the right and two units down, which transformation should be applied?

Accepted Answer

y = (x-3)^2 -2

Answer

y = (x-2)^2 -3

Answer

y = (x+3)^2 +2

Answer

y = (x+2)^2 +3

Question 2

To enclose a rectangular plot of land, 100 feet of fencing is available. What dimensions will maximize the area of this plot?

Accepted Answer

50 feet by 25 feet

Answer

25 feet by 12.5 feet

Answer

75 feet by 12.5 feet

Question 3

Question:
Given f(x) = 1/(x - 2) and g(x) = x^2 + 1, determine the equation of the composite function f(g(x)) and identify any restrictions on its domain.

Accepted Answer

Option 2:
f(g(x)) = 1/(x^2 - 1)
Domain: All real numbers except x = ±1

Answer

Option 1:
f(g(x)) = 1/(x^2 + 1 - 2) = 1/x^2
Domain: All real numbers except x = 0

Answer

Option 3:
f(g(x)) = (x^2 + 1)/(x - 2)
Domain: All real numbers except x = 2

Answer

Option 4:
f(g(x)) = 1/(x^2 + 1)
Domain: All real numbers

Question 4

Analyze the behavior of the function f(x) = (x - 2)/(x + 1) as x approaches infinity and negative infinity. Identify the horizontal and vertical asymptotes of the function.

Accepted Answer

Horizontal asymptote: y = 1; Vertical asymptote: x = -1

Answer

Horizontal asymptote: y = 0; Vertical asymptote: x = 1

Answer

Horizontal asymptote: y = -1; Vertical asymptote: x = 0

Answer

No horizontal asymptote; Vertical asymptote: x = -2

Question 5

Which function represents a rational function?

Accepted Answer

f(x) = (x - 1)/(x + 2)

Answer

f(x) = x² - 2x + 1

Answer

f(x) = eˣ

Answer

f(x) = sin(x)

Question 6

Determine the domain of the function g(x) = √(x - 2).

Accepted Answer

x ≥ 2

Answer

x ≤ 2

Answer

x ∈ R

Answer

x ≠ 2

Question 7

Which transformation shifts the parabola y = x² - 4 four units to the left?

Accepted Answer

f(x) = (x - 4)² - 4

Answer

f(x) = (x + 4)² - 4

Answer

f(x) = x² - 4 + 4

Answer

f(x) = x² - 4 - 4

Question 8

Determine the composition of f(x) = x² and g(x) = x + 1.

Accepted Answer

f(g(x)) = x² + x + 1

Answer

f(g(x)) = x² + 1

Answer

g(f(x)) = x² + 1

Answer

g(f(x)) = x² + x + 1

Question 9

Find the equation of the asymptote for the hyperbola (x - 2)²/9 - (y - 1)²/4 = 1.

Accepted Answer

y = 1

Answer

y = -1

Answer

x = 2

Answer

x = -2

Question 10

Identify the degree of the polynomial 5x⁴ - 2x³ + 1.

Accepted Answer

4

Answer

3

Answer

5

Answer

6

Question 11

Which of the following equations represents a parabola with vertex at (0, 0) and opens upward?

Accepted Answer

y = x^2 + 1

Answer

y = -x^2 + 1

Answer

y = (x + 1)^2

Answer

y = (x - 1)^2

Question 12

Identify the graph of a rational function with a vertical asymptote at x = 2.

Accepted Answer

y = (x - 2)/(x + 2)

Answer

y = (x + 2)/(x - 2)

Answer

y = (2x - 1)/(x + 1)

Answer

y = (2x + 1)/(x - 1)

Question 13

Find the equation of the circle with center (0, 0) and radius 5.

Accepted Answer

x^2 + y^2 = 25

Answer

x^2 + y^2 = 5

Answer

(x + 5)^2 + (y + 5)^2 = 5

Answer

(x - 5)^2 + (y - 5)^2 = 25

Question 14

Which of the following is NOT a property of a polynomial function?

Accepted Answer

Can have a vertical asymptote

Answer

Has a finite number of terms

Answer

Contains only non-negative exponents

Answer

Can cross the x-axis multiple times

Question 15

Which of the following is an example of a polynomial function?

Accepted Answer

y = x^2 + 2x + 1

Answer

y = 1/(x - 1)

Answer

y = sin(x)

Answer

y = e^x

Question 16

Find the composition of functions f(x) = x^2 and g(x) = x + 1.

Accepted Answer

(f ∘ g)(x) = x^2 + 2x + 1

Answer

(f ∘ g)(x) = x^2 + x + 1

Answer

(g ∘ f)(x) = x^4 + 2x^2 + 1

Answer

(g ∘ f)(x) = x^4 + x^2 + 1

Question 17

What is the period of the trigonometric function y = sin(2x)?

Accepted Answer

2π

Answer

π

Answer

π/2

Answer

4π

Question 18

What is the domain of the function f(x) = sqrt(x - 2)?

Accepted Answer

{x | x ≥ 2}

Answer

{x | x ≤ 2}

Answer

{x | x < 2}

Answer

{x | x > 2}

Question 19

What is the vertex form of a quadratic function?

Accepted Answer

f(x) = a(x - h)^2 + k

Answer

f(x) = ax^2 + bx + c

Answer

f(x) = (x - h)(x - k)

Answer

f(x) = ax^3 + bx^2 + cx + d

Question 20

What are the restrictions on the domain of the rational function f(x) = (x - 2)/(x^2 - 4)?

Accepted Answer

The function is undefined at x = -2 and x = 2.

Answer

There are no domain restrictions for this function.

Answer

The function is undefined at x = 2 only.

Answer

The function is undefined at x = 0 and x = 4.

Question 21

What is the inverse function of f(x) = 2x + 3?

Accepted Answer

f^(-1)(x) = (x - 3)/2

Answer

f^(-1)(x) = -2x - 3

Answer

f^(-1)(x) = (x + 3)/2

Answer

f^(-1)(x) = 2x - 3

Question 22

What is the discriminant of a quadratic equation ax^2 + bx + c = 0?

Accepted Answer

b^2 - 4ac

Answer

b^2 + 4ac

Answer

-b^2 + 4ac

Answer

-b^2 - 4ac

Question 23

For the function f(x) = (x - 1)^2 + (y + 2)^2, determine its vertex, axis of symmetry, level curves, and sketch its graph.

Accepted Answer

Vertex: (1, -2)
Axis of Symmetry: x = 1
Level Curves: Circles centered at (1, -2)
Graph: Opens upwards

Answer

Vertex: (2, 1)
Axis of Symmetry: y = 1
Level Curves: Ellipses centered at (2, 1)
Graph: Opens downwards

Answer

Vertex: (-1, -2)
Axis of Symmetry: x = -1
Level Curves: Hyperbolas centered at (-1, -2)
Graph: Opens to the right

Answer

Vertex: (0, 0)
Axis of Symmetry: y = 0
Level Curves: Parabolas opening upwards
Graph: Opens to the left

Question 24

Which transformation shifts the graph of y = f(x) 3 units to the left?

Accepted Answer

y = f(x - 3)

Answer

y = f(x + 3)

Answer

y = f(3x)

Answer

y = f(-3x)

Question 25

Which of the following functions is continuous at all real numbers?

Accepted Answer

f(x) = |x|

Answer

f(x) = 1/x

Answer

f(x) = x² - 4

Answer

f(x) = √x

Question 26

What is the domain of the rational function f(x) = (x - 2)/(x² - 4)?

Accepted Answer

All real numbers except x = 2 and x = -2

Answer

All real numbers except x = 0

Answer

All real numbers

Answer

All real numbers except x = -4

Question 27

Which equation represents the composition of f(x) = x² and g(x) = 2x + 1?

Accepted Answer

g(f(x)) = 4x² + 4x + 1

Answer

f(g(x)) = 4x² + 4x + 1

Answer

f(g(x)) = 2x³ + x

Answer

g(f(x)) = 2x³ + x

Question 28

Which inequality represents the solution set of the system of inequalities y > 3 and x - y < 4?

Accepted Answer

y > 3, x - y < 4

Answer

y > 3, x - y > 4

Answer

y < 3, x - y < 4

Answer

y < 3, x - y > 4

Question 29

For the function f(x) = |x - 2| + |x - 3|, identify the points where the function is not differentiable. Explain your reasoning.

Accepted Answer

x = 2 and x = 3

Answer

x = 0 and x = 1

Answer

The function is differentiable at all real numbers

Question 30

Determine the equation of a parabola that opens downwards with vertex (-3, 2).

Accepted Answer

y = -(x + 3)² + 2

Answer

y = (x - 3)² + 2

Answer

y = -(x - 3)² - 2

Answer

y = (x + 3)² - 2

Question 31

Identify the inverse of the function f(x) = (x - 2)² + 1.

Accepted Answer

f^-1(x) = ±√(x - 1) + 2

Answer

f^-1(x) = (x - 1)² + 2

Answer

f^-1(x) = -√(x - 1) + 2

Answer

f^-1(x) = -√(x + 1) + 2

Question 32

Find the composition of the functions f(x) = x² and g(x) = x + 1.

Accepted Answer

f(g(x)) = x² + x + 1

Answer

f(g(x)) = x³ + x

Answer

f(g(x)) = x² + 1

Answer

f(g(x)) = x + 1

Question 33

What is the equation of the vertical asymptote for the rational function f(x) = (x - 1)/(x² - 4)?

Accepted Answer

x = 2

Answer

x = -2

Answer

y = 2

Answer

y = -2

Question 34

If the graph of the function f(x) = |x| is translated 3 units to the left and 2 units up, what is the equation of the transformed function?

Accepted Answer

g(x) = |x - 3| + 2

Answer

g(x) = |x + 3| + 2

Answer

g(x) = |x - 3| - 2

Answer

g(x) = |x + 3| - 2

Question 35

What is the range of the function f(x) = √(9 - x²)?

Accepted Answer

[0, 3]

Answer

[-3, 3]

Answer

[0, 9]

Answer

[-3, 9]

Question 36

Determine the vertex of the hyperbola defined by the equation (x - 2)²/4 - (y + 1)²/9 = 1.

Accepted Answer

(2, 1)

Answer

(2, -1)

Answer

(-2, 1)

Answer

(-2, -1)

Question 37

Expand and simplify the algebraic expression: (x + 2)(x - 3)(x + 1)

Accepted Answer

x³ - 5x² - 6x - 6

Answer

x³ + 2x² - 5x - 6

Answer

x³ - 2x² + 5x + 6

Answer

x³ + 2x² + 5x + 6

Question 38

Calculate the value of the polynomial function f(x) = x³ - 4x² + 5x - 2 when x = -2

Accepted Answer

-18

Answer

-10

Answer

10

Answer

18

Question 39

Identify the domain of the function f(x) = (x² - 4) / (x + 2)

Accepted Answer

All real numbers except -2

Answer

All real numbers except 2

Answer

All real numbers except 0

Answer

All real numbers

Question 40

Simplify the algebraic expression: (x³y²) / (xy³)

Accepted Answer

x² / y

Answer

y² / x

Answer

x / y

Answer

y / x

Question 41

Which graph represents the equation 3x + 4y = 12?

Accepted Answer

A line with a slope of -3/4

Answer

A line with a y-intercept of 3

Answer

A circle

Answer

A parabola

Question 42

Find the domain of the function f(x) = √(x - 1).

Accepted Answer

[1, ∞)

Answer

[0, ∞)

Answer

(-∞, 0]

Answer

(-∞, 1]

Question 43

Find the zeros of the polynomial function f(x) = x³ - 8.

Accepted Answer

0, 2, -2

Answer

1, -1, 2

Answer

2, -2, 4

Answer

0, 2i, -2i

Question 44

Determine the degree of the polynomial function: f(x) = 3x^5 - 2x^3 + 4x - 1.

Accepted Answer

5

Answer

3

Answer

4

Answer

6

Question 45

Which transformation is applied to the function f(x) to obtain g(x) = f(x - 2) + 3?

Accepted Answer

Horizontal shift of 2 units right and vertical shift of 3 units up

Answer

Vertical shift of 3 units up

Answer

Horizontal shift of 2 units right

Answer

Vertical shift of 2 units up and horizontal shift of 3 units right

Question 46

Determine the composition of functions: f(g(x)) if f(x) = 2x + 1 and g(x) = x^2.

Accepted Answer

4x^2 + 2x + 1

Answer

2x^2 + 1

Answer

x^4 + 2x^2 + 1

Answer

2x^4 + 4x^2 + 1

Question 47

Find the inverse of the function: f(x) = (2x - 1) / (x + 1).

Accepted Answer

f^-1(x) = (x + 1) / (2x - 1)

Answer

f^-1(x) = (2x + 1) / (x - 1)

Answer

f^-1(x) = (x - 1) / (2x + 1)

Answer

f^-1(x) does not exist

Question 48

Which of the following is NOT a characteristic of a rational function?

Accepted Answer

Has a domain that includes all real numbers

Answer

Can be discontinuous at certain points

Answer

Can have vertical asymptotes

Answer

Can be expressed as a ratio of two polynomials

Question 49

Which of the following is an example of a rational function?

Accepted Answer

g(x) = (x - 1)/(x + 2)

Answer

f(x) = x^2 + 2x

Answer

h(x) = e^x + 3

Answer

j(x) = |x - 1|

Question 50

What is the domain of the function f(x) = sqrt(x - 4)?

Accepted Answer

[4, ∞)

Answer

(-∞, 4]

Answer

(-∞, ∞)

Answer

[0, ∞)

Question 51

Which of the following is a factor of the polynomial x^3 + 2x^2 - 5x - 6?

Accepted Answer

x - 3

Answer

x - 1

Answer

x + 2

Answer

x + 6

Question 52

Determine the degree of the polynomial function f(x) = x^3 - 2x^2 + x - 2.

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 53

Find the equation that represents a circle with its center at (h, k) and a radius of r.

Accepted Answer

(x - h)^2 + (y - k)^2 = r

Answer

(x + h)^2 + (y + k)^2 = r

Answer

(x - h)^2 - (y + k)^2 = r

Answer

(x + h)^2 - (y - k)^2 = r

Question 54

Identify the domain of the rational function f(x) = (x - 3) / (x^2 - 9).

Accepted Answer

All real numbers except x = 3 and x = -3

Answer

All real numbers

Answer

All real numbers except x = 0

Answer

The empty set

Question 55

To translate the graph of f(x) 2 units to the left, which transformation should be applied?

Accepted Answer

f(x - 2)

Answer

f(x + 2)

Answer

f(2x)

Answer

f(x/2)

Question 56

Which expression represents the composition of functions?

Accepted Answer

f(g(x))

Answer

g(x) + f(x)

Answer

f(x) - g(x)

Answer

f(x) / g(x)

Question 57

Determine the equation of the asymptote for the function g(x) = (x^2 - 1) / (x - 1).

Accepted Answer

x = 1

Answer

y = 1

Answer

y = x

Answer

x = 0

Question 58

Which of the following is a polynomial function?

Accepted Answer

f(x) = x^3 - 2x + 1

Answer

f(x) = 1 / x

Answer

f(x) = √x

Answer

f(x) = e^x

Question 59

Which of the following geometric figures is a conic section?

Accepted Answer

Circle

Answer

Line

Answer

Triangle

Answer

Square

Question 60

Which of the following functions is a polynomial function?

Accepted Answer

f(x) = 2x^2 + 3x - 1

Answer

g(x) = 1/x

Answer

h(x) = e^x

Answer

j(x) = sin(x)

Question 61

What is the vertex of the parabola y = (x + 2)^2 - 5?

Accepted Answer

(2, -5)

Answer

(-2, -5)

Answer

(-2, 5)

Answer

(2, 5)

Question 62

What is the equation of a horizontal line passing through the point (1, -3)?

Accepted Answer

y = -3

Answer

y = 1

Answer

x = 1

Answer

x = -3

Question 63

What is the leading coefficient of the polynomial f(x) = -2x^3 + 5x^2 - 3x + 1?

Accepted Answer

-2

Answer

5

Answer

-3

Answer

1

Question 64

Determine the discriminant of the quadratic equation 2x^2 - 5x + 3 = 0.

Accepted Answer

11

Answer

-11

Answer

1

Answer

-1