CLEP College Algebra Version 1 Questions

Explanation

<h2>(2, 4) must also lie on the graph of y = f(x).</h2> Given that f'(x) = f(-x), we can infer a symmetry in the function f(x). Since the point (-2, 4) indicates that f(-2) = 4, we also find that f(2) must equal 4 by the property of the function, leading us to conclude that the point (2, 4) lies on the graph. <b>A) (-2,-4)</b> This point does not have to lie on the graph since it contradicts the information given. The point (-2, 4) indicates that f(-2) = 4, so there is no basis for asserting f(-2) = -4. <b>B) (0,0)</b> While (0,0) could be a potential point on some functions, there is no information given that necessitates it to be on the graph. The symmetry condition and the specific point (-2, 4) do not imply that f(0) must equal 0. <b>C) (0,4)</b> Similar to option B, there is no evidence that f(0) must equal 4 based on the information provided. The function's behavior at x = 0 is not determined by the property or the given point. <b>D) (2,-4)</b> This point cannot be on the graph since we established that f(2) = 4 based on the symmetry derived from f'(x) = f(-x). Therefore, there is no support for f(2) being -4. <b>E) (2,4)</b> Since f(-2) = 4 indicates that f(2) must also equal 4, the point (2, 4) is confirmed to lie on the graph of y = f(x). <b>Conclusion</b> The unique property of the function f, defined by f'(x) = f(-x), implies that for every point on the graph, a corresponding symmetric point must also exist. Given the point (-2, 4), we deduced that (2, 4) is a required point on the graph. Other options either contradict the function's symmetry or do not follow logically from the given information.

Explanation

<h2>y = - 3x ^ 2 - 5.</h2> The transformation of the graph begins with a vertical stretch by a factor of 3, followed by an upward translation of 5 units, and finally a reflection across the x-axis. Each transformation modifies the equation step-by-step, leading to the final equation of the graph. <b>A) y = - 3x ^ 2 - 5</b> This equation represents the correct result after applying all the transformations. Starting with y = x^2, stretching it vertically by 3 gives y = 3x^2. Translating this upward by 5 results in y = 3x^2 + 5. Reflecting across the x-axis changes the sign, leading to y = -3x^2 - 5, which matches the final result. <b>B) y = - 3x ^ 2 + 5</b> This option incorrectly reflects the final transformation. While it retains the vertical stretch by a factor of 3, it fails to correctly represent the reflection across the x-axis. The upward translation is also misrepresented, as it should be -5 instead of +5 after reflection. <b>C) y = - 3 * (x - 5) ^ 2</b> This choice misapplies the translation. The expression (x - 5) suggests a horizontal shift to the right by 5 units, which is not part of the described transformations. Although it has the correct vertical stretch and reflection, the translation aspect is incorrect. <b>D) y = - 3 * (x + 5) ^ 2</b> Similar to option C, this equation incorrectly indicates a horizontal shift, this time to the left by 5 units. It maintains the vertical stretch and reflection, but the translation does not align with the operations stated in the question. <b>E) y = 3x ^ 2 – 5</b> This option presents an incorrect reflection across the x-axis and a negative translation. The vertical stretch is present, but the upward translation should not be negative. Therefore, it does not represent the final transformation correctly. <b>Conclusion</b> The transformations applied to the original equation y = x^2 lead to the final equation y = -3x^2 - 5. Each step was essential: the vertical stretch adjusted the steepness, the upward translation shifted the graph, and the reflection across the x-axis inverts the graph. The only equation that accurately reflects these transformations is option A.

Explanation

<h2>-5i</h2> To find the value of 5/i, we can multiply the numerator and denominator by the complex conjugate of the denominator, which is -i. This yields 5/i = 5 * (-i) / (i * -i) = -5i / (-1) = -5i. <b>A) -5</b> This choice is a real number, whereas the computation of 5/i results in a complex number. The value -5 does not represent the result of dividing by the imaginary unit i, therefore it cannot be correct. <b>B) 5i</b> This option suggests a positive imaginary number, but the calculation of 5/i involves multiplying by -i. The result is negative, specifically -5i, making this choice incorrect. <b>C) -5i</b> This is the correct answer as shown in the calculation. When we multiply 5 by the complex conjugate -i and simplify, we arrive at -5i. Thus, this accurately represents the value of 5/i. <b>D) -i/5</b> This choice represents a different complex number. It does not correspond to the computation of 5/i, which produces -5i. Thus, this option fails to match the correct result. <b>E) 5-i</b> This option combines a real and an imaginary component, but does not reflect the result of 5/i. The computation gives a purely imaginary number (-5i), making this choice incorrect. <b>Conclusion</b> In summary, the value of 5/i, when correctly calculated by multiplying by the complex conjugate, simplifies to -5i. Each incorrect choice fails to match this result either by representing a different type of number or by failing to account for the negative sign that arises in the computation. Understanding how to manipulate complex numbers is essential for arriving at the correct solution in this context.

Explanation

<h2>ab is equal to 5 + i.</h2> To find the product of the complex numbers a and b, we multiply (2 + 3i) and (3 - 2i) together. This results in the expression 6 + 4i + 9i - 6i², which simplifies to 5 + i after combining like terms and recalling that i² = -1. <b>A) 0</b> This option suggests that the product of the two complex numbers equals zero. However, the multiplication of (2 + 3i) and (3 - 2i) yields a non-zero result, specifically 5 + i. Therefore, this choice is incorrect. <b>B) - 1 + 5i</b> Calculating the product does not lead to -1 + 5i. The actual computation yields 5 + i, showing that the result is neither negative nor structured as proposed in this option. This choice does not align with the multiplication outcome. <b>C) 1+i</b> This option proposes a product of 1 + i, which is also incorrect. The calculation shows that the correct product is 5 + i, making this option inconsistent with the result of the multiplication. <b>D) 5 + i</b> This is the correct answer. The product of (2 + 3i) and (3 - 2i) simplifies accurately to 5 + i, confirming that this option reflects the correct calculation. <b>E) 5 + 5!</b> This choice erroneously includes the factorial notation, which is irrelevant in this context. The product of two complex numbers does not involve factorials, and thus this expression does not represent the result of the multiplication. <b>Conclusion</b> The product of the complex numbers a and b results in 5 + i, which is confirmed through direct calculation. Each incorrect option fails to match this outcome, either suggesting incorrect values or including extraneous elements that do not apply to the multiplication of complex numbers. Therefore, understanding complex multiplication and simplification is key to arriving at the correct answer.

Explanation

<h2>rt < rs cannot be true if 0 < r < s < t.</h2> In this context, since \( r \), \( s \), and \( t \) are positive and \( r < s < t \), multiplying \( r \) by \( t \) will always yield a smaller product than multiplying \( r \) by \( s \). Therefore, the inequality \( rt < rs \) cannot hold true. <b>A) r ^ 2 < s ^ 2 < t ^ 2</b> This statement is true because squaring each term preserves the order of the inequalities when all variables are positive. Since \( 0 < r < s < t \), squaring each of these values will maintain the same order, confirming that \( r^2 < s^2 < t^2 \). <b>C) 1/t < 1/s < 1/r</b> This statement is also true. Since \( r < s < t \) implies \( 1/t > 1/s > 1/r \) when taking the reciprocals of positive values, the inequalities reverse, confirming \( 1/t < 1/s < 1/r \) as a valid conclusion. <b>D) s/r < t/r</b> This statement can be simplified to \( s < t \) when both sides are multiplied by \( 1/r \) (which is positive), and since \( s < t \) is true under the initial conditions, this inequality holds. <b>E) r/t < r/s</b> This statement can be rearranged to \( s < t \) when both sides are multiplied by \( r \) (which is positive). Since \( s < t \) is indeed true, this inequality is valid. <b>Conclusion</b> In this scenario, the only option that cannot logically hold true, given the conditions \( 0 < r < s < t \), is \( rt < rs \). This inequality contradicts the established order of the variables because multiplying by a larger number (in this case, \( s \) instead of \( t \)) leads to a larger product. All other options respect the relationships among \( r \), \( s \), and \( t \).