Pre-Calculus — Semester A
Free Practice · 10 Questions · 20 min
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Question 1 of 10
TEKS 1A-1G Medium

A quadratic projectile-motion problem produces two algebraic solutions: t = 2 seconds and t = −4 seconds. The problem asks when the ball hits the ground after being hit. Which response demonstrates the BEST mathematical reasoning?

A Reject t = 2 because the smaller solution is always the answer
B Average the two: t = (2 + (−4))/2 = −1 second
C Reject t = −4 (negative time has no physical meaning here); report t = 2 seconds
D Report both because they are algebraically valid
Explanation
A complete solution requires checking that the algebraic answer fits the physical context. Both t = 2 and t = −4 are algebraic roots, but negative time has no physical meaning when the timer starts at t = 0 — so only t = 2 is contextually valid. Common mistakes: (C) there's no rule that smaller is always correct; (D) averaging roots gives the parabola's axis of symmetry, which is irrelevant to "when does it hit the ground."
Question 2 of 10
TEKS 3E-3I Easy Diagram
-3 -2 -1 1 2 3 4 -1 1 2 3 4 5 x y(0, 1)

The graph below shows an exponential function passing through (0, 1), growing rapidly to the right, and approaching 0 as x → −∞. Which function is shown?

A f(x) = 2^x
B f(x) = −2^x
C f(x) = log₂(x)
D f(x) = x²
Explanation
Exponential growth f(x) = a^x with a > 1: passes through (0, 1), increases rapidly for positive x, approaches 0 (never touches) as x → −∞. (B) ln/log has opposite shape; (C) negates everything; (D) parabola has symmetry about y-axis, doesn't approach 0.
Question 3 of 10
TEKS 3E-3I Easy

Simplify log₂(8) + log₂(4) using a single logarithm and evaluate.

A log₂(12) ≈ 3.58
B log₂(64) = 6
C log₂(2) = 1
D log₂(32) = 5
Explanation
Product rule for logarithms: log_b(M) + log_b(N) = log_b(M · N) — when the bases match, addition combines into a log of the product (NOT the sum). So log₂(8) + log₂(4) = log₂(32). Since 2⁵ = 32, the value is 5. Common mistake (A): adding arguments (8 + 4 = 12) instead of multiplying. Verify directly: log₂8 = 3 and log₂4 = 2, so 3 + 2 = 5 ✓.
Question 4 of 10
TEKS 2A-2J Medium

Given f(x) = x² + 1 and g(x) = 2x − 3, find (f ∘ g)(2).

A 2
B 9
C 0
D 5
Explanation
Composition (f ∘ g)(x) = f(g(x)) — apply g first, then f. Step 1: g(2) = 2(2) − 3 = 1. Step 2: f(1) = 1² + 1 = 2. Common mistake (C): computing (g ∘ f)(2) instead — f(2) = 5. Order matters: in f ∘ g, the function written second (g) is applied first.
Question 5 of 10
TEKS 2A-2J Hard

A piecewise function is defined as:

f(x) = x + 2 if x < 1
f(x) = 3x if x ≥ 1

Find f(1) and determine whether f is continuous at x = 1.

A f(1) = 3, and f IS continuous because the left-hand limit (1+2=3) equals the function value (3·1=3)
B f(1) = 1 + 2 = 3, but the function jumps to 3·1 = 3 from the right (which is the same), so it's continuous by coincidence only
C f(1) = 3, and f is NOT continuous because piecewise functions are never continuous at their boundaries
D f(1) is undefined because x = 1 is on the boundary
Explanation
x = 1 falls into the second piece (x ≥ 1), so f(1) = 3(1) = 3. Continuity check: left-hand limit from the first piece: lim(x→1⁻) (x + 2) = 3. Right-hand limit and value: f(1) = 3. Since all three agree, f is continuous at x = 1. Key rule: a piecewise function is continuous at a boundary iff both one-sided limits agree with the function's value there. (A) is a common myth — piecewise functions CAN be continuous when pieces match.
Question 6 of 10
TEKS 3A-3D Easy

Find all REAL zeros of P(x) = x³ − 4x² + x − 4 by factoring.

A x = 4, x = i, x = −i (treating i as real)
B x = −4 only
C x = 4, x = 1, x = −1
D x = 4 only
Explanation
Factor by grouping: P(x) = x²(x − 4) + 1(x − 4) = (x − 4)(x² + 1). Solving each factor: x − 4 = 0 → x = 4; x² + 1 = 0 → x² = −1, which has NO real solutions (only complex i and −i). The question asks for REAL zeros, so the answer is x = 4 only. Verify: P(4) = 64 − 64 + 4 − 4 = 0 ✓.
Question 7 of 10
TEKS 3E-3I Medium

A bank account pays 6% annual interest compounded continuously. If $5,000 is deposited, the balance after t years is A(t) = 5000 · e^(0.06t). To the nearest dollar, what is the balance after 10 years?

A $8,000
B $50,000
C $5,600
D $9,111
Explanation
Continuous compounding: A = P·e^(rt). Here P = $5,000, r = 0.06, t = 10. Compute: 0.06·10 = 0.6, then e^0.6 ≈ 1.8221, then 5000·1.8221 ≈ $9,111. Common mistakes: (A) uses a flawed simple-interest estimate; (B) round guess; (D) multiplies principal by t (confuses compound growth with linear growth). Insight: even moderate rates over long horizons nearly double the principal due to the exponential nature of e^(rt).
Question 8 of 10
TEKS 3A-3D Medium

Identify the vertical asymptote and any holes of R(x) = (x² − 4) / (x² − 5x + 6).

A No vertical asymptote; holes at x = 2 and x = 3
B VA at x = 3; hole at x = 2
C VAs at x = 2 and x = 3; no holes
D VA at x = 2; hole at x = 3
Explanation
Factor both: x² − 4 = (x−2)(x+2); x² − 5x + 6 = (x−2)(x−3). Simplify: R(x) = (x+2)/(x−3), with the restriction x ≠ 2. The cancelled (x−2) leaves a hole at x = 2 (function undefined there, but simplified form is). The remaining (x−3) in the denominator gives a vertical asymptote at x = 3 (simplified form blows up). Rule: cancelled factors → holes; uncancelled denominator factors → vertical asymptotes.
Question 9 of 10
TEKS 1A-1G Easy

Three students each model the same growth pattern 1, 2, 4, 8, 16, …

Student A writes f(n) = 2^(n−1). Student B writes f(1) = 1, f(n) = 2·f(n−1). Student C draws a table. Which statement is TRUE?

A Student C is wrong because tables can never replace a formula
B All three correctly represent the same pattern in valid forms
C Only Student B is correct because recursive forms are more rigorous
D Only Student A is correct because formulas are required
Explanation
Multiple representations of the same function — explicit formula, recursive rule, table, or graph — are all valid mathematical communication (TEKS 1.D–1.E). Student A's explicit form gives f(1)=1, f(2)=2, f(3)=4, … ✓. Student B's recursive rule generates the same sequence step by step. A complete table also encodes the pattern. Choose whichever representation makes the problem easiest.
Question 10 of 10
TEKS 2A-2J Easy

Find the domain of f(x) = √(x − 4).

A (−∞, ∞)
B (4, ∞)
C [4, ∞)
D (−∞, 4)
Explanation
The expression inside a square root must be ≥ 0 for a real output. So x − 4 ≥ 0 → x ≥ 4 → interval [4, ∞). The bracket [ at 4 includes x = 4 because √(4 − 4) = √0 = 0 is a real number. Common mistake (D): writing (4, ∞) excludes 4, but √0 is defined. Use parentheses only when the endpoint is NOT in the domain.

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