Rasmussen MAT 121 College Algebra Final Exam 2026/2027 |
200 Verified Questions & Answers | Instant Download
Q1. What is the solution set for the linear equation 3x - 7 = 5x + 9?
A. x = -8
B. x = 2
C. x = 8
D. x = -2
Correct Answer: A. x = -8
Rationale: Subtract 3x both sides -7 =2x+9 -> -16=2x -> x=-8. Common error sign flip gives 8. Solve step by step.
Q2. What is the slope of the line passing through points (2, 5) and (6, 13)?
A. 0.5
B. 2
C. 8
D. -2
Correct Answer: B. 2
Rationale: Slope m=(y2-y1)/(x2-x1)=(13-5)/(6-2)=8/4=2. 8 is numerator only. 0.5 is reciprocal. -2 sign error.
Q3. What is the equation of the line with slope 3 and y-intercept -4 in slope-intercept form?
A. y = 3x + 4
B. y = -4x + 3
C. y = 3x - 4
D. y = -3x -4
Correct Answer: C. y = 3x - 4
Rationale: y=mx+b m=3 b=-4 => y=3x-4. Swap m and b gives -4x+3. Sign error +4.
Q4. What is the domain of the function f(x) = sqrt(x - 2)?
A. x <= 2
B. All real numbers
C. x > 2 or (2, infinity)
D. x >= 2 or [2, infinity)
Correct Answer: D. x >= 2 or [2, infinity)
Rationale: Square root radicand >=0 x-2>=0 x>=2 inclusive [2,inf). >2 excludes 2 incorrectly.
Q5. What is the range of f(x)=x^2 -4?
A. y >= -4 or [-4, infinity)
B. All real numbers
C. y <= -4
D. y >= 4
Correct Answer: A. y >= -4 or [-4, infinity)
Rationale: Parabola vertex (0,-4) opens up minimum -4 range y>=-4.
Q6. What are the x-intercepts of y = x^2 - 9?
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A. (0,-3) and (0,3)
B. (-3,0) and (3,0)
C. (-3,0) only
D. (-9,0) and (9,0)
Correct Answer: B. (-3,0) and (3,0)
Rationale: x-intercepts y=0 x^2=9 x=+-3 so (-3,0),(3,0).
Q7. What is the vertex of the parabola y = (x - 2)^2 + 3?
A. (2,-3)
B. (-2,-3)
C. (2,3)
D. (-2,3)
Correct Answer: C. (2,3)
Rationale: Vertex form y=a(x-h)^2+k vertex (h,k) h=2 k=3 => (2,3). Sign confusion -2.
Q8. What is the solution to x^2 +5x+6=0 by factoring?
A. x=2, x=-3
B. x=2, x=3
C. x=-2, x=3
D. x=-2, x=-3
Correct Answer: D. x=-2, x=-3
Rationale: Factor (x+2)(x+3)=0 => x=-2,-3.
Q9. What is the discriminant of 2x^2 -4x +1=0 and number of real solutions?
A. 8, two real solutions
B. 16, two real
C. -8, no real solutions
D. 0, one real solution
Correct Answer: A. 8, two real solutions
Rationale: Discriminant b^2-4ac = (-4)^2-4*2*1=16-8=8 >0 two real.
Q10. What is the simplified form of (3x^2 y^3)(4x^4 y)?
A. 12x^6 y^3
B. 12x^6 y^4
C. 12x^8 y^3
D. 7x^6 y^4
Correct Answer: B. 12x^6 y^4
Rationale: Multiply coefficients 3*4=12 add exponents x^2+4=6 y^3+1=4 =>12x^6 y^4.
Q11. What is the result of (2x-3)(x+4)?
A. 2x^2+8x-12
B. 2x^2-12
C. 2x^2+5x-12
D. 2x^2-5x-12
Correct Answer: C. 2x^2+5x-12
Rationale: FOIL 2x*x=2x^2 2x*4=8x -3*x=-3x -3*4=-12 =>2x^2+5x-12.
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Q12. What is the factored form of 9x^2 -16?
A. (3x-4)^2
B. (3x-16)(3x+16)
C. (9x-4)(9x+4)
D. (3x-4)(3x+4)
Correct Answer: D. (3x-4)(3x+4)
Rationale: Difference of squares a^2-b^2 (3x)^2-4^2 => (3x-4)(3x+4).
Q13. What is the solution to 2/(x) = 4/(x+2)?
A. x=2
B. x=-2
C. x=4
D. x=0
Correct Answer: A. x=2
Rationale: Cross multiply 2(x+2)=4x =>2x+4=4x =>4=2x =>x=2. x=-2 extraneous denominator zero.
Q14. What is the inverse function of f(x)=2x+6?
A. f^-1(x)=(x+6)/2
B. f^-1(x)=(x-6)/2
C. f^-1(x)=x/2+6
D. f^-1(x)=2x-6
Correct Answer: B. f^-1(x)=(x-6)/2
Rationale: y=2x+6 swap x=2y+6 => x-6=2y => y=(x-6)/2.
Q15. What is f(g(x)) if f(x)=x^2 and g(x)=x+3?
A. x^2+3
B. x^2+9
C. (x+3)^2 = x^2+6x+9
D. x+3^2
Correct Answer: C. (x+3)^2 = x^2+6x+9
Rationale: Composition f(g)= (x+3)^2 expand x^2+6x+9.
Q16. Which function is one-to-one and passes horizontal line test?
A. f(x)=x^2
B. f(x)=x^2+3
C. f(x)=|x|
D. f(x)=2x+1 linear
Correct Answer: D. f(x)=2x+1 linear
Rationale: Linear with non-zero slope one-to-one horizontal line test passes. x^2 and |x| fail.
Q17. What is the horizontal asymptote of f(x)=(3x^2+2)/(x^2-1)?
A. y=3
B. No horizontal asymptote
C. y=1
D. y=0
Correct Answer: A. y=3
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Rationale: Degree numerator = denominator ratio leading coefficients 3/1=3 => y=3.
Q18. What is the vertical asymptote of f(x)=1/(x-5)?
A. y=5
B. x=5
C. x=-5
D. x=0
Correct Answer: B. x=5
Rationale: Denominator zero x-5=0 => x=5 vertical asymptote.
Q19. What is the solution to |2x-3|=7?
A. x=-2 only
B. x=5 only
C. x=5, x=-2
D. x=7, x=-7
Correct Answer: C. x=5, x=-2
Rationale: Absolute 2x-3=7 =>2x=10 x=5 and 2x-3=-7 =>2x=-4 x=-2.
Q20. What is the center and radius of (x-1)^2+(y+2)^2=9?
A. Center (-1,2) radius 9
B. Center (1,2) radius 3
C. Center (1,-2) radius 9
D. Center (1,-2) radius 3
Correct Answer: D. Center (1,-2) radius 3
Rationale: Standard circle (x-h)^2+(y-k)^2=r^2 h=1 k=-2 r=sqrt9=3.
Q21. What is the solution to system y=x+1 and y=2x-1 by substitution?
A. (2,3)
B. (3,2)
C. (0,1)
D. (1,2)
Correct Answer: A. (2,3)
Rationale: Set x+1=2x-1 =>1+1=2x-x =>2=x then y=3 => (2,3).
Q22. What is the y-intercept of 4x+2y=8?
A. (2,0)
B. (0,4)
C. (0,2)
D. (0,8)
Correct Answer: B. (0,4)
Rationale: Set x=0 =>2y=8 y=4 => (0,4). (2,0) is x-intercept.
Q23. What is the exponential form of log2(8)=3?
A. 8^2=3
B. 2^8=3
C. 2^3=8
D. 3^2=8
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200 Verified Questions & Answers | Instant Download
Q1. What is the solution set for the linear equation 3x - 7 = 5x + 9?
A. x = -8
B. x = 2
C. x = 8
D. x = -2
Correct Answer: A. x = -8
Rationale: Subtract 3x both sides -7 =2x+9 -> -16=2x -> x=-8. Common error sign flip gives 8. Solve step by step.
Q2. What is the slope of the line passing through points (2, 5) and (6, 13)?
A. 0.5
B. 2
C. 8
D. -2
Correct Answer: B. 2
Rationale: Slope m=(y2-y1)/(x2-x1)=(13-5)/(6-2)=8/4=2. 8 is numerator only. 0.5 is reciprocal. -2 sign error.
Q3. What is the equation of the line with slope 3 and y-intercept -4 in slope-intercept form?
A. y = 3x + 4
B. y = -4x + 3
C. y = 3x - 4
D. y = -3x -4
Correct Answer: C. y = 3x - 4
Rationale: y=mx+b m=3 b=-4 => y=3x-4. Swap m and b gives -4x+3. Sign error +4.
Q4. What is the domain of the function f(x) = sqrt(x - 2)?
A. x <= 2
B. All real numbers
C. x > 2 or (2, infinity)
D. x >= 2 or [2, infinity)
Correct Answer: D. x >= 2 or [2, infinity)
Rationale: Square root radicand >=0 x-2>=0 x>=2 inclusive [2,inf). >2 excludes 2 incorrectly.
Q5. What is the range of f(x)=x^2 -4?
A. y >= -4 or [-4, infinity)
B. All real numbers
C. y <= -4
D. y >= 4
Correct Answer: A. y >= -4 or [-4, infinity)
Rationale: Parabola vertex (0,-4) opens up minimum -4 range y>=-4.
Q6. What are the x-intercepts of y = x^2 - 9?
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A. (0,-3) and (0,3)
B. (-3,0) and (3,0)
C. (-3,0) only
D. (-9,0) and (9,0)
Correct Answer: B. (-3,0) and (3,0)
Rationale: x-intercepts y=0 x^2=9 x=+-3 so (-3,0),(3,0).
Q7. What is the vertex of the parabola y = (x - 2)^2 + 3?
A. (2,-3)
B. (-2,-3)
C. (2,3)
D. (-2,3)
Correct Answer: C. (2,3)
Rationale: Vertex form y=a(x-h)^2+k vertex (h,k) h=2 k=3 => (2,3). Sign confusion -2.
Q8. What is the solution to x^2 +5x+6=0 by factoring?
A. x=2, x=-3
B. x=2, x=3
C. x=-2, x=3
D. x=-2, x=-3
Correct Answer: D. x=-2, x=-3
Rationale: Factor (x+2)(x+3)=0 => x=-2,-3.
Q9. What is the discriminant of 2x^2 -4x +1=0 and number of real solutions?
A. 8, two real solutions
B. 16, two real
C. -8, no real solutions
D. 0, one real solution
Correct Answer: A. 8, two real solutions
Rationale: Discriminant b^2-4ac = (-4)^2-4*2*1=16-8=8 >0 two real.
Q10. What is the simplified form of (3x^2 y^3)(4x^4 y)?
A. 12x^6 y^3
B. 12x^6 y^4
C. 12x^8 y^3
D. 7x^6 y^4
Correct Answer: B. 12x^6 y^4
Rationale: Multiply coefficients 3*4=12 add exponents x^2+4=6 y^3+1=4 =>12x^6 y^4.
Q11. What is the result of (2x-3)(x+4)?
A. 2x^2+8x-12
B. 2x^2-12
C. 2x^2+5x-12
D. 2x^2-5x-12
Correct Answer: C. 2x^2+5x-12
Rationale: FOIL 2x*x=2x^2 2x*4=8x -3*x=-3x -3*4=-12 =>2x^2+5x-12.
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Q12. What is the factored form of 9x^2 -16?
A. (3x-4)^2
B. (3x-16)(3x+16)
C. (9x-4)(9x+4)
D. (3x-4)(3x+4)
Correct Answer: D. (3x-4)(3x+4)
Rationale: Difference of squares a^2-b^2 (3x)^2-4^2 => (3x-4)(3x+4).
Q13. What is the solution to 2/(x) = 4/(x+2)?
A. x=2
B. x=-2
C. x=4
D. x=0
Correct Answer: A. x=2
Rationale: Cross multiply 2(x+2)=4x =>2x+4=4x =>4=2x =>x=2. x=-2 extraneous denominator zero.
Q14. What is the inverse function of f(x)=2x+6?
A. f^-1(x)=(x+6)/2
B. f^-1(x)=(x-6)/2
C. f^-1(x)=x/2+6
D. f^-1(x)=2x-6
Correct Answer: B. f^-1(x)=(x-6)/2
Rationale: y=2x+6 swap x=2y+6 => x-6=2y => y=(x-6)/2.
Q15. What is f(g(x)) if f(x)=x^2 and g(x)=x+3?
A. x^2+3
B. x^2+9
C. (x+3)^2 = x^2+6x+9
D. x+3^2
Correct Answer: C. (x+3)^2 = x^2+6x+9
Rationale: Composition f(g)= (x+3)^2 expand x^2+6x+9.
Q16. Which function is one-to-one and passes horizontal line test?
A. f(x)=x^2
B. f(x)=x^2+3
C. f(x)=|x|
D. f(x)=2x+1 linear
Correct Answer: D. f(x)=2x+1 linear
Rationale: Linear with non-zero slope one-to-one horizontal line test passes. x^2 and |x| fail.
Q17. What is the horizontal asymptote of f(x)=(3x^2+2)/(x^2-1)?
A. y=3
B. No horizontal asymptote
C. y=1
D. y=0
Correct Answer: A. y=3
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Rationale: Degree numerator = denominator ratio leading coefficients 3/1=3 => y=3.
Q18. What is the vertical asymptote of f(x)=1/(x-5)?
A. y=5
B. x=5
C. x=-5
D. x=0
Correct Answer: B. x=5
Rationale: Denominator zero x-5=0 => x=5 vertical asymptote.
Q19. What is the solution to |2x-3|=7?
A. x=-2 only
B. x=5 only
C. x=5, x=-2
D. x=7, x=-7
Correct Answer: C. x=5, x=-2
Rationale: Absolute 2x-3=7 =>2x=10 x=5 and 2x-3=-7 =>2x=-4 x=-2.
Q20. What is the center and radius of (x-1)^2+(y+2)^2=9?
A. Center (-1,2) radius 9
B. Center (1,2) radius 3
C. Center (1,-2) radius 9
D. Center (1,-2) radius 3
Correct Answer: D. Center (1,-2) radius 3
Rationale: Standard circle (x-h)^2+(y-k)^2=r^2 h=1 k=-2 r=sqrt9=3.
Q21. What is the solution to system y=x+1 and y=2x-1 by substitution?
A. (2,3)
B. (3,2)
C. (0,1)
D. (1,2)
Correct Answer: A. (2,3)
Rationale: Set x+1=2x-1 =>1+1=2x-x =>2=x then y=3 => (2,3).
Q22. What is the y-intercept of 4x+2y=8?
A. (2,0)
B. (0,4)
C. (0,2)
D. (0,8)
Correct Answer: B. (0,4)
Rationale: Set x=0 =>2y=8 y=4 => (0,4). (2,0) is x-intercept.
Q23. What is the exponential form of log2(8)=3?
A. 8^2=3
B. 2^8=3
C. 2^3=8
D. 3^2=8
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