Quiz 5 Spring 2016 Math 107 College Algebra University Of Ma

Quiz 5 Spring 2016 Math 107 College Algebrauniversity Of Marylan

Find the degree, the leading term, the leading coefficient, the constant term and the end behavior of the given polynomial: f(x) = 4x^5 - x^2 + 2x + 5

Find the domain of the following rational function. Write it in the form p(x)/q(x) for polynomial functions p and q and simplify: f(x) = 1 - 3x + 1

Sketch a detailed graph of: f(x) = 2x / (x^2 - 1)

Perform the indicated operation and simplify: 10x(x - 3)^(-1) + 5x^2(-1)(x - 3)^(-2)

Solve the rational equation: 2x / (x + 3) = 5

Let f(x) = x^2 and g(x) = 1 + √x. Find and simplify the indicated composite function. Also, state the domain of the composite function: (f ◦ g)(x)

Graph the function and use the Horizontal Line Test to check if the function is one-to-one: f(x) = x^2 - 2x + 2

Perform the indicated operations and simplify: 3 * √64x^{14}

For the following function, state its domain and create a sign diagram: f(x) = x * √(x - 1)

Simplify the following: log [expression not specified in the original prompt]

Paper For Above instruction

The provided set of problems from Spring 2016 Math 107 College Algebra at the University of Maryland encompasses a broad spectrum of fundamental algebraic concepts, including polynomial analysis, rational functions, graphing, solving equations, composite functions, and logarithmic simplification. This comprehensive overview aims to address each problem systematically, elucidating key principles and demonstrating detailed approaches consistent with college-level algebra instruction.

Problem 1: Polynomial Analysis

The polynomial in question is f(x) = 4x^5 - x^2 + 2x + 5. To analyze its degree, leading term, leading coefficient, constant term, and end behavior, proceed as follows:

• The degree of a polynomial is the highest power of x with a non-zero coefficient. Here, the highest power is 5, so the degree is 5.
• The leading term is the term with the highest degree, which is 4x^5.
• The leading coefficient is the coefficient of the leading term, which is 4.
• The constant term is the term without x, which is 5.
• End behavior for a polynomial with an odd degree and a positive leading coefficient is that as x approaches positive infinity, f(x) approaches positive infinity, and as x approaches negative infinity, f(x) approaches negative infinity.

Thus, the polynomial's end behavior indicates it rises to infinity on the right and falls to negative infinity on the left, typical for degree 5 with a positive leading coefficient.

Problem 2: Domain of Rational Function

The function is f(x) = 1 - 3x + 1, which simplifies to f(x) = 2 - 3x; however, the original problem indicates a rational function, suggesting an error or missing information. Assuming the intended rational function is f(x) = 1/(x - 1), the domain excludes values that make the denominator zero.

• Simplify the expression if necessary; if the original expression was f(x) = (1 - 3x + 1), then it is not rational. If it is a typo, and the intended function is f(x) = 1/(x - 1), then
• The domain is all real numbers except x = 1 (where the denominator is zero).

Therefore, the domain is all real x such that x ≠ 1.

Problem 3: Graph of f(x) = 2x / (x^2 - 1)

The function involves rational expressions with denominators that can cause vertical asymptotes where the denominator is zero. Since x^2 - 1 = (x - 1)(x + 1), the asymptotes are at x = 1 and x = -1.

To graph:

• Identify vertical asymptotes at x = -1 and x = 1.
• Check the end behavior: as x → ±∞, f(x) → 0.
• Analyze the sign of the numerator and denominator in each interval determined by the asymptotes.
• Plot key points and asymptotes to visualize the behavior, noting that the graph approaches but does not cross the asymptotes.

Problem 4: Simplify the Expression

Given: 10x(x - 3)^(-1) + 5x^2 (-1) (x - 3)^(-2). Rewrite as:

• First term is 10x / (x - 3).
• Second term simplifies to -5x^2 / (x - 3)^2.

Expressed over a common denominator (x - 3)^2, the sum becomes:

(10x)(x - 3) / (x - 3)^2 - 5x^2 / (x - 3)^2

Simplify numerator: 10x(x - 3) = 10x^2 - 30x

Combine over common denominator:

(10x^2 - 30x - 5x^2) / (x - 3)^2 = (5x^2 - 30x) / (x - 3)^2

Factor numerator: 5x(x - 6)

Final simplified form: 5x(x - 6) / (x - 3)^2.

Problem 5: Solving Rational Equation

Equation: 2x / (x + 3) = 5

• Multiply both sides by (x + 3): 2x = 5(x + 3)
• Distribute: 2x = 5x + 15
• Bring all to one side: 2x - 5x = 15
• Simplify: -3x = 15
• Divide both sides by -3: x = -5

Check that the solution does not make the denominator zero; since x ≠ -3, x = -5 is valid.

Problem 6: Composite Function and Domain

Given f(x) = x^2 and g(x) = 1 + √x. Compute (f ◦ g)(x):

• (f ◦ g)(x) = f(g(x)) = (g(x))^2 = (1 + √x)^2
• Expand: (1)^2 + 2 1 √x + (√x)^2 = 1 + 2√x + x

Domain considerations:

• √x is real only when x ≥ 0
• Since the expression involves √x, the domain is x ≥ 0

The composite function is: (f ◦ g)(x) = 1 + 2√x + x, with x ≥ 0.

Problem 7: Graphing and Horizontal Line Test

Function: f(x) = x^2 - 2x + 2

• This quadratic opens upward with vertex at x = -b/(2a) = 2/(2) = 1.
• Vertex coordinates: (1, f(1)) = (1, 1 - 2 + 2) = (1, 1)
• Graph is a parabola with vertex at (1, 1).
• To check if function is one-to-one, use the Horizontal Line Test: Since the parabola opens upward and passes through (1, 1), horizontal lines above the vertex intersect twice, indicating the function is not one-to-one.

Problem 8: Simplify the Expression

3 * √(64x^{14})

• √(64x^{14}) = √64 √x^{14} = 8 x^{7}
• Multiplying by 3: 3 8 x^{7} = 24x^{7}

Final result: 24x^7.

Problem 9: Domain and Sign Diagram

Function: f(x) = x * √(x - 1)

• Domain: x - 1 ≥ 0 ⇒ x ≥ 1

Sign analysis:

• For x ≥ 1: f(x) is non-negative since x ≥ 1 and √(x - 1) ≥ 0.
• At x = 1: f(1) = 1 * √0 = 0.
• As x increases beyond 1, both x and √(x - 1) are positive, so f(x) > 0.

Problem 10: Logarithmic Simplification

The specific expression to simplify was not fully provided in the prompt. Typically, logarithmic properties such as product, quotient, and power rules are used.

• If the expression involves logs of multiplication: log(a * b) = log a + log b.
• If division: log(a / b) = log a - log b.
• If powers: log(a^k) = k * log a.

Therefore, without the explicit expression, a general approach would involve applying these properties accordingly once the expression is specified.

References

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