MAC 1147 Final Exam Review Spring 2024

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Questions and Answers

What is the vertex of the hyperbola 25x^2 - 100y^2 = 1?

(10, 0)

(0, 0) (correct)

(0, 5)

(5, 0)

What is the value of cot(π/3)?

1/√3 (correct)

1/3

√3

What is the value of sin(𝛼 + 𝛽) if sin 𝛼 = 5/13 and cos 𝛽 = 12/13?

5/13 (correct)

12/13

-1

What is the domain of f(x) = log₉(x - 6)?

(6, ∞) Signup and view all the answers

What is the equation of the graph of f(x) = -x^2(x - 1)(x + 3)?

Cubic function with a negative leading coefficient Signup and view all the answers

What is the amplitude of the function y = 2cos(2x + π)?

2 Signup and view all the answers

What is the value of sin²(x) + cos²(x) cot²(x)?

1 Signup and view all the answers

What is the phase shift of the function y = -2sin(4x - π)?

-π/4 Signup and view all the answers

What is the inverse of the function f(x) = (7x + 6) / (x - 5)?

f⁻¹(x) = (5x - 6) / (x - 7) Signup and view all the answers

What is the value of log₇√(x⁵y / 49) simplified?

(1/2)log₇(x) + log₇(y) - 1 Signup and view all the answers

What is the period of the function y = 2cos(2x + 4)?

π Signup and view all the answers

What is the value of cos(θ) given tan(θ) = -3 and θ in quadrant 2?

-1/√10 Signup and view all the answers

What is the value of tan(π)?

0 Signup and view all the answers

What is the value of cos(3π/2)?

0 Signup and view all the answers

What is the horizontal asymptote of h(x) = (10x³) / (2x² + 1)?

y = 10 Signup and view all the answers

What is the general term of the arithmetic sequence with a1 = -19 and d = 7?

an = -19 + 7(n - 1) Signup and view all the answers

What is the value of the expression 5log₇(4) + log₇(r - 8) - log₇r?

log₇(20/r) Signup and view all the answers

What is the measure of the angle between the end of the shadow and the vertical side of the building?

tan⁻¹(2.6) Signup and view all the answers

What is the inverse of the one-to-one function f(x) = (2x + 2) / (x - 2)?

f⁻¹(x) = (2x - 2) / (x + 2) Signup and view all the answers

What is the domain of the function f(x) = x³ - 36x?

All real numbers Signup and view all the answers

What is the value of (g ∘ f)(-3)?

4 Signup and view all the answers

What is the domain of the function g(x) = x² - 9 / (x + 2)?

All real numbers except -2 Signup and view all the answers

What is the length of the guy wire that makes an angle of 20° with the ground?

246 feet Signup and view all the answers

What is the exact value of tan⁻¹(√3)?

60° Signup and view all the answers

What is the value of (f ◦ g)(1) using the graphs of f and g?

-2 Signup and view all the answers

What is the solution to the logarithmic equation log 4 (x − 4) + log 4(x − 10) = 2?

8 Signup and view all the answers

What is the domain of the function g(x) = √(x-9)/√(x-1)?

x > 1 and x ≠ 9 Signup and view all the answers

What is the exact value of sin⁻¹(√2/2)?

π/4 Signup and view all the answers

What is the value of f(-1) for the given piecewise function?

-7 Signup and view all the answers

What is the polynomial with zeros -8, 1 (multiplicity 2), and degree 3?

x³ - 2x² - 7x + 8 Signup and view all the answers

What is the exact value of sec(−3π/4)?

-√2 Signup and view all the answers

What is the domain of the function f(x) = √(14-x)/x²?

x > 0 and x < 14 Signup and view all the answers

What is the solution of the equation 5𝑥 + 6 / 22𝑥 - 7?

𝑥 = 2 Signup and view all the answers

What is the inverse of 𝑓(𝑥) = 3𝑥 - 2?

𝑓^(-1)(𝑥) = (𝑥 + 2) / 3 Signup and view all the answers

What is the value of 𝜃 if 𝜃 = tan^(-1) (13/5)?

𝜃 = 69° Signup and view all the answers

What is the range of the function 𝑓(𝑥) = |𝑥| - 3?

[3, ∞) Signup and view all the answers

What is the amplitude, period, and phase shift of the function 𝑓(𝑥) = 2sin(𝑥 + 𝜋/8)?

Amplitude = 2, Period = 2𝜋, Phase shift = -𝜋/8 Signup and view all the answers

What is the value of 𝑎𝑛 in the sequence 𝑎𝑛 = 7𝑛 - 26?

𝑎𝑛 = 7𝑛 - 26 Signup and view all the answers

What is the domain of the function 𝑓(𝑥) = 1 / (𝑥 - 3)?

(-∞, 3) ∪ (3, ∞) Signup and view all the answers

What is the 𝑥-intercept of the function 𝑓(𝑥) = 𝑥^4 - 3𝑥^3 + 2𝑥^2 - 12𝑥?

(-3, 0) and (1, 0) Signup and view all the answers

Study Notes

Composite Functions

Evaluate composite functions using graphs, e.g., (f ◦ g)(1)

Logarithmic Equations

Solve logarithmic equations, e.g., log 4 (x − 4) + log 4 (x − 10) = 2
Reject any value not in the domain of the original logarithmic expressions

Function Domain

Find the domain of functions, e.g., g(x) = √(x−9)
Domain may be a union of intervals, e.g., (1, 9) ∪ (9, ∞)

Trigonometry

Solve triangles, e.g., 7, 30°, and 115°
Evaluate trig functions, e.g., sin−1 (2)
Use reference angles to find exact values, e.g., sec (4π/3)

Polynomials

Form a polynomial with given zeros and degree, e.g., (x + 8)(x - 1)^2

Piecewise Functions

Evaluate piecewise functions, e.g., f(-1)
Graph piecewise functions, e.g., f(x) = {4x - 3 if x < -1, -2x + 4 if x ≥ -1}

Inverse Functions

Find the inverse of a one-to-one function, e.g., f(x) = x/(x - 5)
Evaluate inverse functions, e.g., f(-2), f(0), f(2)

Logarithmic Expressions

Expand logarithmic expressions using properties of logarithms, e.g., log 7 (√(x^5 y) / 49)
Evaluate logarithmic expressions, e.g., log (x + 30) - log (3) = log (2x + 1)

Function Operations

Evaluate the difference quotient for a function, e.g., f(x) = x^2 + 9x - 2

Graphing

Graph functions, e.g., 4x^2 = 64 - 16y^2
Identify vertices, foci, and asymptotes of conic sections, e.g., ellipse
Analyze graphs of functions, e.g., f(x) = -x^2 (x - 1)(x + 3)

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Description

This quiz reviews composite functions and graph analysis for MAC 1147 final exam in Spring 2024. It covers evaluating composite functions using graphs of f and g.