CLEP College Algebra Version 2 Questions

Explanation

<h2>(2, 4)</h2> The function f is even, as indicated by the property f(x) = f(-x) for all real numbers x. This symmetry means that if the point (-2, 4) is on the graph of y = f(x), then the corresponding point (2, 4) must also be present due to the nature of even functions. <b>A) (-2, -4)</b> This point does not satisfy the property of the function. Since f(-2) = f(2) and we know f(-2) = 4, it follows that f(2) must also equal 4, not -4. Therefore, (-2, -4) cannot be on the graph. <b>B) (0, 0)</b> While the point (0, 0) could lie on the graph of an even function, there is no given information to confirm its presence. The function could take on any value at x = 0, and without specific information about f(0), we cannot assert that (0, 0) is on the graph. <b>C) (0, 4)</b> Similar to (0, 0), the point (0, 4) may or may not be on the graph. The evenness of the function does not imply that f(0) equals 4 unless explicitly stated; hence we cannot conclude its presence on the graph. <b>D) (2, -4)</b> This point contradicts the even function property. Since f(-2) = 4, it follows that f(2) = 4 as well. Therefore, (2, -4) cannot be a valid point on the graph of y = f(x). <b>E) (2, 4)</b> This point must lie on the graph, as established by the even function property. Since the function is symmetric and f(-2) = 4, it follows that f(2) must also equal 4, confirming that (2, 4) is indeed on the graph. <b>Conclusion</b> The even nature of the function f ensures that for any point (a, b) on the graph, the point (-a, b) is also present. Given that (-2, 4) is on the graph, the corresponding point (2, 4) must also be included, while the other options either contradict the function's properties or lack sufficient information to confirm their inclusion.

Explanation

<h2>z is equal to (9-7i).</h2> To find the value of z, we multiply the complex numbers (3+i) and (2-3i), resulting in z = 9 - 7i. This involves applying the distributive property and combining like terms, specifically paying attention to the multiplication of imaginary units. <b>A) (3-7i)</b> This choice incorrectly represents the result of the multiplication. It suggests a different real part (3) which does not align with the calculation, as the correct real part from (3+i)(2-3i) is 9. <b>B) (3+5i)</b> This option is not correct because it suggests an entirely different outcome, with both real and imaginary parts not matching the computed result. The values of both the real (3) and imaginary (5) components are inconsistent with the multiplication performed. <b>C) (6-6i)</b> This answer does not correspond to the correct calculation as it presents incorrect values for both the real and imaginary parts. The result of the multiplication yields a real part of 9 and an imaginary part of -7, which are not reflected here. <b>D) (9-7i)</b> This choice accurately reflects the computed result of the multiplication of the two complex numbers, where the real part is 9 and the imaginary part is -7, confirming the correct evaluation of z. <b>E) (9+5i)</b> This option is incorrect because, while the real part (9) matches, the imaginary part (5) does not correspond with the result of the multiplication. The correct imaginary component should be -7, not +5. <b>Conclusion</b> The complex number z, found through the multiplication of (3+i) and (2-3i), evaluates to (9-7i). The only option that accurately reflects this result is D. Understanding the multiplication of complex numbers is essential for correctly determining their sum and product, highlighting the importance of both the real and imaginary components in the final answer.

Explanation

<h2>The sum shown is consistent with n/(n+1).</h2> This sum can be derived from the properties of series and sequences, particularly when considering the limit behavior of fractions as n approaches infinity. The expression n/(n+1) simplifies nicely to 1 when n is very large, aligning with the expected behavior of the sum. <b>A) 1/n</b> The expression 1/n approaches zero as n increases, which does not reflect the behavior of the sum in question. The sum is not consistent with 1/n, as this would imply diminishing returns rather than a converging value approaching 1. <b>B) n/(n+1</b> This expression simplifies to 1 - 1/(n+1) and approaches 1 as n becomes very large. This matches the behavior of the sum, making it the correct answer, as it reflects the limit that the series converges to. <b>C) (n+1)/(n+2)</b> The expression (n+1)/(n+2) approaches 1 as well, but more slowly than n/(n+1). The difference in growth rates indicates that while both may converge to similar limits, the sum specifically aligns better with n/(n+1) due to its more precise relationship with the terms involved. <b>D) 1/(n+1)</b> The expression 1/(n+1) also approaches zero as n increases, similar to 1/n. It does not represent a constant value or convergence to 1, making it inconsistent with the behavior of the sum in question. <b>Conclusion</b> The sum in question aligns with the expression n/(n+1), which reflects the limit behavior thoroughly, approaching 1 as n increases. Other options either approach zero or converge at different rates, failing to match the intended behavior of the sum. Understanding these relationships is crucial for grasping the underlying concepts of series and limits in mathematical analysis.

Explanation

<h2>F(m) - T(m) is defined by the expression 7m + 3/(m + 1).</h2> To find the difference between the frog and turtle populations, we calculate F(m) - T(m) using the provided functions F(m) = m + 6 and T(m) = (m^2 + 3)/(m + 1). Simplifying this expression results in 7m + 3/(m + 1). <b>A) 7m + 9/(m + 1)</b> This expression incorrectly adds 9 instead of 3 to the numerator when simplifying the difference between the two populations. The correct calculation must account for the original functions, leading to a different numerator. <b>B) -7m - 3/(m + 1)</b> This choice incorrectly presents the difference as negative, suggesting that the turtle population is always larger than the frog population. The functions indicate that the frog population increases linearly, while the turtle population grows quadratically, leading to a different conclusion. <b>C) 7m + 3/(m + 1)</b> This is the correct expression for F(m) - T(m). It accurately represents the difference in the populations by subtracting the turtle population function from the frog population function, resulting in a linear term and a fraction that reflects the growth rates of both populations. <b>D) #NAME?</b> This option indicates an error in the expression, likely resulting from a misinterpretation or typographical error in the function representation. It does not provide a valid mathematical expression for the population difference. <b>E) #NAME?</b> Similar to option D, this choice also presents an error, failing to define a valid expression for the difference between the frog and turtle populations, which prevents it from being a viable answer. <b>Conclusion</b> The difference between the frog and turtle populations is accurately represented by the expression 7m + 3/(m + 1). This calculation illustrates the growth patterns of both species over time, emphasizing the linear growth of the frog population in contrast to the quadratic growth of the turtle population. The incorrect options either miscalculate or misrepresent the necessary population dynamics, highlighting the importance of careful function manipulation in determining differences.

Explanation

<h2>The equation (x-1)(x^2+x+1)(x^2-3x-4)=0 has exactly three distinct real roots.</h2> To find the distinct real roots of the equation, we analyze each factor. The factor (x-1) provides one distinct real root at x = 1. The quadratic (x^2 + x + 1) has no real roots, as its discriminant is negative. The quadratic (x^2 - 3x - 4) can be factored into (x-4)(x+1), yielding two distinct real roots at x = 4 and x = -1. Therefore, the total number of distinct real roots is three. <b>A) One</b> This option suggests that the equation has only one distinct real root. However, the factor (x-1) gives one root, while the other factor (x^2 - 3x - 4) contributes two additional distinct real roots, contradicting the assertion that there is only one root. <b>B) Two</b> Choosing two distinct real roots ignores the contribution from (x-1) and the factored form of (x^2 - 3x - 4), which reveals two additional roots. Thus, the total number of distinct real roots is three, making this option incorrect. <b>C) Three</b> This is the correct option since we have one root from (x-1) and two roots from (x^2 - 3x - 4). The quadratic (x^2 + x + 1) does not introduce any real roots, confirming a total of three distinct real roots. <b>D) Four</b> This incorrect choice implies there are four distinct real roots. However, the analysis reveals only three distinct roots from the factors. The factor (x^2 + x + 1) does not contribute any real roots, resulting in a total less than four. <b>E) Five</b> Claiming five distinct real roots is incorrect as well. The analysis shows only three distinct roots in total. Each quadratic factor and linear factor has been accounted for, confirming that five distinct real roots do not exist in this equation. <b>Conclusion</b> The equation (x-1)(x^2+x+1)(x^2-3x-4)=0 has a total of three distinct real roots: one from (x-1) and two from the factorization of (x^2 - 3x - 4). The absence of real roots from (x^2 + x + 1) further solidifies the conclusion that the total count of distinct real roots is three, validating option C as the correct answer.