Mixed Review of Problem Solving Answers Algebra 2
A set of algebra ii problems with their detailed solutions to cocky test and diagnose your background and review and gain deep agreement on the following topics:
Complex Numbers
Trouble ane-1
Permit z = ii - 3 i where i is the imaginary unit. Evaluate z z* , where z* is the conjugate of z , and write the answer in standard grade.
Detailed Solution.
Problem 1-ii
Evaluate and write in standard grade \( \dfrac{1-i}{2-i} \) , where i is the imaginary unit.
Detailed Solution.
Quadratic Equations
Problem 2-i
Detect all solutions of the equation \( x(10 + iii) = - five \).
Detailed Solution.
Problem 2-two
Discover all values of the parameter m for which the equation \( -2 x^two + m x = ii m \) has circuitous solutions.
Detailed Solution.
Functions
Trouble 3-1
Allow \( f(10) = - ten^two + 3(10 - 1) \). Evaluate and simplify \( f(a-ane)\).
Detailed Solution.
Problem iii-2
Write, in interval notation, the domain of function \(f\) given by \(f(x) = \sqrt{x^two-16} \).
Detailed Solution.
Problem 3-iii
Find and write, in interval annotation, the range of function \(f\) given by \(f(x) = - 10^two - 2x + 6 \).
Detailed Solution.
Trouble three-4
Let \(f(ten) = \sqrt{10 - 2} \) and \(g(ten) = x^ii + 2 \); evaluate \( (f_o g)(a - 1) \) for \( a \lt i \).
Detailed Solution.
Problem 3-five
Which of the following is a 1-to-ane function?(At that place may be more 1 answer).
a) \(f(x) = - 2 \) b) \(g(x) = \ln(x^2 - i) \) c) \(h(x) = |x| + 2 \) d) \(j(ten) = 1/x + 2 \) due east) \(k(x) = \sin(10) + 2 \) f) \(fifty(x) = ln(x - 1) + 1 \)
Detailed Solution.
Problem 3-6
What is the inverse of function f given past \(f(x) = \dfrac{-10+ii}{x-1}\)?
Detailed Solution.
Problem iii-7
Allocate the following functions as even, odd or neither.
a) \(f(x) = - ten^iii \) b) \(chiliad(x) = |x|+ ii \) c) h(x) = \( \ln(x - one) \)
Detailed Solution.
Problem 3-8
Function \(f \) has one nada only at \(10 = -two\). What is the zero of the function \(2f(2x - 5) \)?
Detailed Solution.
Problem iii-9
Which of the post-obit piecewise functions has the graph shown below?
a) \( f(x) = \brainstorm{cases} 10^2 & \text{if} \; x \ge 0 \\ 2 & \text{if} \; -2 \lt 10 \lt 0\\ - ten + one& \text{if} \; x \le -two \end{cases} \) b) \( g(x) = \begin{cases} 10^2 & \text{if} \; x \gt 0 \\ two & \text{if} \; -2 \lt x \le 0\\ - x + 1& \text{if} \; x \le -2 \stop{cases} \) c) \( h(x) = \begin{cases} x^2 & \text{if} \; ten \gt 0 \\ ii & \text{if} \; -ii \lt ten \lt 0\\ - 10 + 1 & \text{if} \; x \lt -2 \terminate{cases} \)
Detailed Solution.
Trouble 3-10
Summate the average rate of change of function \( f(x) = \dfrac{1}{x} \) as 10 changes from \( x = a\) to \( x = a + h \).
Detailed Solution.
Polynomials
Problem 4-1
Find the caliber and the remainder of the division \( \dfrac{-ten^4+2x^3-ten^ii+5}{x^2-2} \).
Detailed Solution.
Problem four-2
Find \( k \) and so that the remainder of the partition \( \dfrac{4 10^2+2x-3}{2 x + k} \) is equal to \( -1 \)?
Detailed Solution.
Trouble 4-iii
\( (x - 2) \) is one of the factors of \( p(x) = -2x^4-8x^3+2x^2+32x+24 \). Cistron \(p\) completely.
Detailed Solution.
Problem 4-4
Factor \( 16 10^4 - 81 \) completely.
Detailed Solution.
Problem 4-5
Find all solutions to the equation \( (ten - three)(ten^2 - iv) = (- ten + 3)(10^2 + 2x) \)
Detailed Solution.
Problem four-vi
Solve the inequality \( (x + ii)(10^2-4x-5) \ge (-x - two)(10+one)(10-3)\)
Detailed Solution.
Problem 4-7
The graph of a polynomial part is shown below. Which of the following functions can possibly have this graph?
a) \( y = -(x+two)^five(x-i)^ii \) b) \( y = 0.5(x+2)^3(10-1)^2 \) c) \( y = -0.v(x+2)^3 (x-i)^2 \) d) \( y = -(10+two)^3(10-1)^ii \)
Detailed Solution.
Problem 4-8
Which of the post-obit graphs could possibly exist that of the function f given by \( f(ten) = k (x - 1)(x^2 + iv) \) where k is a negative constant? Find m if possible.
Detailed Solution.
Rational Expressions, Equations, Inequalities and Functions
Problem 5-1
Write equally a unmarried rational expression: \( \dfrac{ten^2+3x-5}{(x-one)(10+two)} - \dfrac{two}{x+2} - 1 \).
Detailed Solution.
Problem v-2
Solve the equation: \( \dfrac{- 10^two+v}{x-ane} = \dfrac{ten-2}{x+two} - 4 \).
Detailed Solution.
Problem 5-3
Solve the inequality: \( \dfrac{i}{x-i}+\dfrac{one}{x+1} \ge \dfrac{3}{x^2-one} \).
Detailed Solution.
Trouble 5-4
Notice the horizontal and vertical asymptotes of the part: \( y = \dfrac{3x^ii}{5 x^2 - two ten - vii} + ii \).
Detailed Solution.
Problem 5-five
Which of the following rational functions has an oblique asymptote? Find the point of intersection of the oblique asymptote with the function.
a) \( y = -\dfrac{x-1}{x^ii+2} \) b) \( y = -\dfrac{x^4-1}{x^ii+2} \) c) \( y = -\dfrac{10^3 + 2x ^ 2 -i}{ten^2- 2} \) d) \( y = -\dfrac{x^2-i}{10^2+2} \)
Detailed Solution.
Trouble five-6
Which of the post-obit graphs could be that of role \( f(x) = \dfrac{2x-two}{x-one} \)?
Detailed Solution.
Trigonometry and Trigonometric Functions
Trouble 6-1
A rotating wheel completes m rotations per minute. Determine the athwart speed of the wheel in radians per 2d.
Detailed Solution.
Problem six-2
Determine the verbal value of \( sec(-xi\pi/3) \).
Detailed Solution.
Problem 6-3
Catechumen 1200� in radians giving the verbal value.
Detailed Solution.
Problem 6-iv
Catechumen \( \dfrac{-7\pi}{9} \) in degrees giving the exact value.
Detailed Solution.
Problem 6-5
What is the range and the period of the the role \( f(10) = -two\sin(-0.5(x - \pi/v)) - half dozen \)?
Detailed Solution.
Problem half-dozen-6
Which of the following graphs could be that of function given by: \( y = - \cos(2x - \pi/iv) + 2 \)?
Detailed Solution.
Trouble vi-7
Find a possible equation of the grade \( y = a \sin(b x + c) + d \) for the graph shown below.(there are many possible solutions)
Detailed Solution.
Problem 6-8
Find the smallest positive value of x, in radians, such that \( - 4 \cos (2x - \pi/four) + one = 3 \)
Detailed Solution.
Problem 6-9
Simplify the expression: \( \dfrac{\cot(ten)\sin(x) + \cos(x) \sin^2(ten)+\cos^iii(x)}{\cos(10)} \)
Detailed Solution.
Logarithmic and Exponential Functions
Problem 7-1
Simplify the expression \( \dfrac{4x^ii y^8}{8 x^iii y^v} \) using positive exponents in the final answer.
Detailed Solution.
Problem vii-2
Evaluate the expression \( \dfrac{three^{1/three} ix^{i/three}}{4^{one/2}} \).
Detailed Solution.
Problem 7-3
Rewrite the expression \( \log_b(2x - 4) = c \) in exponential form.
Detailed Solution.
Problem 7-4
Simplify the expressiomn: \( \log_a(9) \cdot \log_3(a^ii) \)
Detailed Solution.
Trouble 7-5
Solve the equation \( \log(x + 1) - log(10 - 1) = 2 \log(x + one) \).
Detailed Solution.
Problem seven-6
Solve the equation \( e^{2x} + e^10 = half dozen \).
Detailed Solution.
Problem 7-7
What is the horizontal asymptote of the graph of \( f(x) = 2 ( - two - east^{10-1}) \)?
Detailed Solution.
Problem 7-8
What is the vertical asymptote of the graph of \( f(x) = log(2x - half dozen) + 3 \)?
Detailed Solution.
Problem 7-9
Match the given functions with the graph shown beneath?
A) \( y = ii - 0.v^{2x-1} \) B) \( y = 0.5^{2x-ane} \) C) \( y = two - 0.5^{-2x+1} \) D) \( y = 0.v^{-2x+1} \)
Detailed Solution.
Problem 7-10
Match the given functions with the graph shown below?
A) \( y = two+ln(x-2) \) B) \( y=-log_2(x+1)-ane \) C) \( y = -ln(-x) \) D) \( y = y=-log_3(ten+ane)-one \)
Detailed Solution.
Source: https://www.analyzemath.com/Algebra2/Algebra-2.html
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