College algebra assignments

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UNIT TWO

UNIVERSITY OF THE PEOPLE

1201-01: COLLEGE ALGEBRA




TASK 1: FUNCTION OPERATIONS

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Given functions:


f(x) = 2x + 1


g(x) = 3x + 1

(i) FUNCTIONAL VALUES OF OPERATIONS

1. Addition (f+g)(x)

(f+g)(x) = f(x) + g(x) = (2x+1) + (3x+1) = 5x + 2

2. Multiplication (fg)(x)

(fg)(x) = f(x) × g(x) = (2x+1)(3x+1) = 6x² + 2x + 3x + 1 = 6x² + 5x + 1

3. Composition fog(x)

fog(x) = f(g(x)) = f(3x+1) = 2(3x+1) + 1 = 6x + 2 + 1 = 6x + 3

4. Composition gof(x)

gof(x) = g(f(x)) = g(2x+1) = 3(2x+1) + 1 = 6x + 3 + 1 = 6x + 4

(ii) EQUALITY OF OPERATIONS

No, fg, fog, and gof are not equal:

· (fg)(x) = 6x² + 5x + 1 (quadratic function)

· fog(x) = 6x + 3 (linear function)

· gof(x) = 6x + 4 (linear function with different constant)

Only fog and gof are both linear functions, but they have different constant terms (3 vs 4), so they are not

equal either. Function composition is generally not commutative.

(iii) DOMAINS AND RANGES

1. Domain of (f+g)(x): All real numbers (R) since both f and g are linear polynomials.

Range: All real numbers (R) since 5x+2 is linear.

2. Domain of (fg)(x): All real numbers (R) since both functions are polynomials.

Range: Since this is a quadratic opening upward with vertex at x = -5/12, the minimum value is 6(-5/12)²

+ 5(-5/12) + 1 = -1/24. Range: [-1/24, ∞)