#1
Which of the following is a linear function?
y = 2x + 3
ExplanationLinear functions have a constant rate of change.
#2
Solve for x: 2x + 5 = 13
x = 6
ExplanationIsolate x by subtracting 5 from both sides, then divide by 2.
#3
If f(x) = 3x - 2, find f(4).
10
ExplanationReplace x with 4 in the function and evaluate.
#4
What is the domain of the function g(x) = sqrt(2x - 5)?
x > 5/2
ExplanationThe expression under the square root cannot be negative.
#5
Simplify the expression: (2x^2 - 5x + 3) / (x - 1)
2x + 3
ExplanationUse polynomial long division or factorization.
#6
If k(x) = |2x - 7|, what is the range of k?
k ≥ 0
ExplanationAbsolute value functions output non-negative values.
#7
Solve the inequality: 2x - 7 > 3x + 1
x < -8
ExplanationIsolate x by subtracting 2x from both sides.
#8
Find the sum of the roots of the quadratic equation: x² - 6x + 9 = 0.
3
ExplanationSum of roots for a quadratic ax² + bx + c = 0 is -b/a.
#9
If q(x) = 4x² - 16, what is the vertex of the parabola represented by q?
(4, 0)
ExplanationThe vertex of the parabola is at (-b/2a, q(-b/2a)).
#10
Solve the system of equations:
2x + 3y = 8
4x - y = 6
x = 2, y = 1
ExplanationUse substitution or elimination method to solve.
#11
If r(x) = 5x³ + 2x² - 3x + 1, what is the degree of the polynomial r?
3
ExplanationThe degree of the polynomial is the highest power of x.
#12
Solve the inequality: 3(x - 2) < 2x + 5
x > 3
ExplanationDistribute and solve for x.
#13
If u(x) = sin(2x), what is the period of the function u?
2π
ExplanationThe period of sin(kx) is 2π/k.
#14
Simplify the expression: (4x^3 - 2x^2 + 7x - 1) / (2x^2 - 3x + 1)
2x + 1
ExplanationPerform polynomial division or factorization.
#15
If h(x) = (x + 1)/(x - 3), find the vertical asymptote.
x = 3
ExplanationVertical asymptotes occur where the denominator equals zero.
#16
Evaluate the expression: log₂(16) + log₂(8)
6
ExplanationCombine logarithms using the property log(a) + log(b) = log(ab).
#17
If p(x) = x³ - 4x² + 5x - 2, find the critical points of p.
x = 1, x = 3
ExplanationCritical points occur where the derivative equals zero.
#18
Find the inverse function of f(x) = 2x + 4.
f⁻¹(x) = (x - 4)/2
ExplanationSwap x and y, then solve for y.
#19
If m(x) = e^(3x), what is the derivative of m with respect to x?
3e^(3x)
ExplanationUse the chain rule: d(e^u)/dx = u'(x)e^u.
#20
Evaluate the limit as x approaches 2 for the function: f(x) = (x² - 4) / (x - 2)
6
ExplanationTry direct substitution or factorization.
#21
If n(x) = 2^(4x), what is the logarithmic form of n?
log₂(2^4)
ExplanationConvert exponentiation to logarithmic form.
#22
Find the derivative of the function g(x) = ln(x^2 - 1) with respect to x.
(2x)/(x^2 + 1)
ExplanationUse the chain rule and the derivative of ln(u).
#23
Evaluate the integral: ∫(3x² + 2x + 1) dx
x³ + x² + x + C
ExplanationIntegrate each term separately.
#24
Find the solution to the equation: |2x - 1| = 5
x = 3/2
ExplanationSet both positive and negative expressions equal to the given value.
#25
If v(x) = e^(-x), what is the limit as x approaches infinity for v?
0
ExplanationAs x approaches infinity, e^(-x) approaches 0.