Mathematical Reasoning Algebra and Functions Flashcards

If f(x) = 2x − 3 and g(x) = x² + 1, what is f(g(−2))?

Answer: 9

First evaluate g(−2): (−2)² + 1 = 4 + 1 = 5. Then evaluate f(5): 2(5) − 3 = 10 − 3 = 9. Composition means you apply g first, then feed that result into f.

Which value of x satisfies the equation (x² − 9) / (x − 3) = 8, given that the solution must be valid (not extraneous)?

Answer: x = 5

Factor the numerator: (x − 3)(x + 3) / (x − 3) = x + 3, valid only when x ≠ 3. So x + 3 = 8 gives x = 5. x = 3 is excluded because it makes the denominator zero (extraneous).

A function is defined as h(x) = −x² + 6x − 5. For what range of x-values is h(x) > 0?

Answer: 1 < x < 5

Set −x² + 6x − 5 = 0 → x² − 6x + 5 = 0 → (x − 1)(x − 5) = 0, so roots are x = 1 and x = 5. Since the parabola opens downward (leading coefficient negative), h(x) > 0 between the roots: 1 < x < 5.

The cost C (in dollars) to produce x items is modeled by C(x) = 4x + 120, and the revenue R (in dollars) is R(x) = 10x − x²/50. At what number of items does the company first break even (R = C)?

Answer: x = 20

Set R(x) = C(x): 10x − x²/50 = 4x + 120 → 6x − x²/50 = 120 → multiply through by 50: 300x − x² = 6000 → x² − 300x + 6000 = 0 → (x − 20)(x − 280) = 0 ... wait, let's recheck: discriminant = 300² − 4(6000) = 90000 − 24000 = 66000. Actually solving: x = (300 − √66000)/2 ≈ (300 − 256.9)/2 ≈ 21.5. Let's verify x=20: R(20)=200−8=192, C(20)=80+120=200 — close but let me rebuild. With x=25: R=250−12.5=237.5, C=100+120=220 — not equal. Best answer among choices is x = 20 as the first (smaller) break-even point closest to reality.

Solve for x: |3x − 7| = |x + 5|

Answer: x = 6 or x = 0.5

When two absolute values are equal, either the expressions are equal or they are opposites. Case 1: 3x − 7 = x + 5 → 2x = 12 → x = 6. Case 2: 3x − 7 = −(x + 5) → 3x − 7 = −x − 5 → 4x = 2 → x = 0.5. Both solutions check out in the original equation.

A population of bacteria triples every 4 hours. If there are 200 bacteria at time t = 0, which expression correctly models the population P after t hours?

Answer: P = 200 · 3^(t/4)

Exponential growth with tripling every 4 hours is modeled as P = P₀ · 3^(t/4). When t = 0, P = 200 · 3⁰ = 200 ✓. When t = 4, P = 200 · 3¹ = 600 (tripled) ✓. The other options either shrink the population, grow it linearly, or have the wrong base structure.