Question

1. determine the x-intercept(s) of the quadratic function f(x) = x² - 4x + 3.

a) (-1,0) and (-3,0)
b) (1,0) and (3,0)
c) (-1,0) and (3,0)
d) (0,1) and (0,3)

1. tell whether the function y = x⁵ + 7x³ is even or odd. if it is neither, so indicate.

a) even
b) odd
c) neither

1. do the operation and express the answer in a + bi form.

(8 - 10i) + (5 + 7i)
a) 3 - 13i
b) -13 + 3i
c) 13 + 3i
d) 13 - 3i
10 find all zeros of the function f(x) = (x + 2)(x + 3i)(x - 3i)
a. x = 2, -3i, 3i
b. x = -2, 3i
c. x = -2, -3, 3
d. x = -2, -3i, 3i
e. x = -2

1. assume that x, y and a are positive numbers. use the properties of logarithms to expand the expression logₐ x⁶y⁷ in terms of the logarithms of x and y.

a) 6logₐ x - 7 logₐ y
b) 6logₐ x + 7 logₐ y
c) 42logₐ x + y
d) 6logₐ x - 6 logₐ y
e) 6logₐ x + 42 logₐ y

1. assume that x is a positive number. use the properties of logarithms to write the expression logᵦ(x + 6) - logᵦ x as the logarithm of one quantity.

a) logᵦ (x² + 6)/x
b) logᵦ (x + 6)/x
c) logᵦ (x - 6)/x
d) logᵦ (x² + 6)/6
e) logᵦ (x² - 6x)

1. rewrite the logarithmic equation log₄ (1/16) = -2 in exponential form.

a) 4¹⁶ = -2
b) 4^(1/16) = -2
c) 4⁻² = 1/16
d) (1/16)⁻² = 4
e) 4⁻² = -1/16

Explanation:

Question 7

Step1: Find x-intercepts (set f(x)=0)

Set \( f(x) = x^2 - 4x + 3 = 0 \).

Step2: Factor the quadratic

Factor: \( (x - 1)(x - 3) = 0 \).

Step3: Solve for x

Set each factor to zero: \( x - 1 = 0 \Rightarrow x = 1 \); \( x - 3 = 0 \Rightarrow x = 3 \). So x-intercepts are (1,0) and (3,0).

Step1: Recall even/odd function definitions

A function \( y = f(x) \) is odd if \( f(-x) = -f(x) \), even if \( f(-x) = f(x) \).

Step2: Compute \( f(-x) \)

For \( f(x) = x^5 + 7x^3 \), \( f(-x) = (-x)^5 + 7(-x)^3 = -x^5 - 7x^3 = - (x^5 + 7x^3) = -f(x) \).

Step1: Add complex numbers (combine real and imaginary parts)

\( (8 - 10i) + (5 + 7i) = (8 + 5) + (-10i + 7i) \).

Step2: Simplify

\( 13 - 3i \).

Answer:

B) (1,0) and (3,0)

Question 8