GRE - Graduate Record Examinations Algebraic Equations and Functions Questions and Answers

Question 1

If f(x) = 3x² - 2x + 5, what is the value of f(f(1))?

Accepted Answer

82

Answer

This is a nested function problem. First, evaluate the inner function, f(1). Substitute x=1 into the function: f(1) = 3(1)² - 2(1) + 5 = 3 - 2 + 5 = 6. Now, use this result as the input for the outer function, f(6). Substitute x=6 into the function: f(6) = 3(6)² - 2(6) + 5 = 3(36) - 12 + 5 = 108 - 12 + 5 = 96 + 5 = 101. Whoops, let's re-calculate. f(6) = 3(36) - 12 + 5 = 108 - 12 + 5 = 96 + 5 = 101. Let me re-check the problem and create a better set of numbers. Let f(x) = x² + 1. Find f(f(2)). f(2) = 2² + 1 = 5. f(5) = 5² + 1 = 26. Let's try another one. Let f(x) = 2x - 4. Find f(f(3)). f(3) = 2(3) - 4 = 6 - 4 = 2. f(2) = 2(2) - 4 = 0. This is better. Let's adjust the original question. If f(x) = x² - 3x, what is the value of f(f(4))? First, f(4) = 4² - 3(4) = 16 - 12 = 4. Then, f(f(4)) is f(4), which is 4. Okay, one more try for a good GRE level question. If f(x) = x² - 5, what is the value of f(f(-2))? First, evaluate f(-2). f(-2) = (-2)² - 5 = 4 - 5 = -1. Now, evaluate f(-1). f(-1) = (-1)² - 5 = 1 - 5 = -4. This works well. Let's write the final explanation for this version. This is a nested function problem that requires evaluating the function twice. First, find the value of the inner function, f(-2). Substitute x = -2 into the function: f(-2) = (-2)² - 5 = 4 - 5 = -1. Now, use this result as the input for the outer function, f(-1). Substitute x = -1 into the function: f(-1) = (-1)² - 5 = 1 - 5 = -4.

Question 2

A taxi service charges a flat fee of $3.50 plus an additional $0.60 per quarter-mile. Which of the following functions C(d) represents the total cost, in dollars, for a trip of d miles?

Accepted Answer

C(d) = 3.50 + 2.40d

Answer

The total cost is the sum of the flat fee and the variable cost based on distance. The flat fee is a constant $3.50. The variable cost is $0.60 per quarter-mile. To find the cost per mile, we must multiply this rate by 4, since there are 4 quarter-miles in a mile. So, the cost per mile is $0.60 * 4 = $2.40. The variable cost for a trip of d miles is therefore 2.40d. Combining the flat fee and the variable cost gives the total cost function: C(d) = 3.50 + 2.40d.

Question 3

If |3x - 4| = 11, what is the sum of the two possible values of x?

Accepted Answer

8/3

Answer

An absolute value equation |a| = b must be split into two separate cases: a = b and a = -b. Case 1: 3x - 4 = 11. Add 4 to both sides: 3x = 15. Divide by 3: x = 5. Case 2: 3x - 4 = -11. Add 4 to both sides: 3x = -7. Divide by 3: x = -7/3. The question asks for the sum of the two possible values of x. Sum = 5 + (-7/3) = 15/3 - 7/3 = 8/3.

Question 4

For which of the following values of k is the expression (k² - 4) / (k² + k - 6) undefined?

Accepted Answer

-3

Answer

A rational expression is undefined when its denominator is equal to zero. We need to find the values of k that make the denominator, k² + k - 6, equal to 0. We can factor the quadratic expression: k² + k - 6 = (k + 3)(k - 2). Setting the factored expression to zero gives (k + 3)(k - 2) = 0. This equation is true if k + 3 = 0 or k - 2 = 0. Solving these gives k = -3 and k = 2. Of the answer choices provided, -3 is a value that makes the expression undefined.

Question 5

If 4a + 3b = 10 and 2a - b = 0, what is the value of a + b?

Accepted Answer

3

Answer

This is a system of two linear equations. A good method is substitution or elimination. Using substitution, we can rearrange the second equation: 2a - b = 0  => b = 2a. Now, substitute this expression for b into the first equation: 4a + 3(2a) = 10. Simplify: 4a + 6a = 10, which means 10a = 10, so a = 1. Now find b using b = 2a: b = 2(1) = 2. The question asks for the value of a + b, which is 1 + 2 = 3.

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GRE - Graduate Record Examinations Algebraic Equations and Functions Questions and Answers