Question

Which of the following statements have the same result? Explain each step in solving each one.

I. f(1) when f(x) = 4x - 1
II. f⁻¹ (2) when f(x) = 3x-1/4
III. 2y + 8 = 4y - 2

a) I and II only
b) I and III only
c) II and III only
d) All of the above

Answer 1

Upon evaluating each statement, Statement I results in 3, Statement II gives an input for the inverse function corresponding to the output of 2, and Statement III results in y being 5. None of the statements yield the same result, hence the answer is none of the above.

Step-by-step explanation:

The question asks which of three mathematical statements have the same result. First, let's evaluate each statement:

1. I. To find f(1) when f(x) = 4x - 1, we simply substitute x with 1, which gives us f(1) = 4(1) - 1 = 3.
2. II. To find f⁻¹(2) when f(x) = (3x - 1)/4, we need to solve for x in the equation 3x - 1 = 4 * 2. This simplifies to 3x = 9, hence x = 3. This result is the input that would give an output of 2 in the original function; therefore, this is f⁻¹(2).
3. III. The equation 2y + 8 = 4y - 2 is a linear equation that we can solve for y. Subtracting 2y from both sides gives 8 + 2 = 2y. Dividing both sides by 2 yields y = 5.

Comparing the results, Statement I yields a result of 3, Statement II gives us the input corresponding to 2 for the inverse function, and Statement III results in the value of y being 5. Therefore, the statements that have the same result are none of the above.