Problem 51 - Algebra 2

Question

For which function does \(f\left(-\frac{1}{2}\right) \neq-1 ?\) A. \(f(x)=2 x\) B. \(f(x)=|-2 x|\) C. \(f(x)=[x]\) D. \(f(x)=[2 x]\)

Step-by-Step Solution

Answer

Option B: \(f(x) = |-2x|\) satisfies the condition.

Step 1: Evaluate Option A

Let's substitute \(x = -\frac{1}{2}\) into the first function, \(f(x) = 2x\).\[ f\left(-\frac{1}{2}\right) = 2 \times \left(-\frac{1}{2}\right) = -1 \]Since \(f\left(-\frac{1}{2}\right) = -1\), this option does not satisfy the condition.

Step 2: Evaluate Option B

Now, substitute \(x = -\frac{1}{2}\) into the second function, \(f(x) = |-2x|\).\[ f\left(-\frac{1}{2}\right) = | -2 \times \left(-\frac{1}{2}\right)| = |1| = 1 \]Since \(f\left(-\frac{1}{2}\right) = 1\) and \(1 eq -1\), Option B satisfies the condition.

Step 3: Evaluate Option C

Substitute \(x = -\frac{1}{2}\) into the third function, \(f(x) = [x]\), where \([x]\) denotes the floor function.\[ f\left(-\frac{1}{2}\right) = \left\lfloor -\frac{1}{2} \right\rfloor = -1 \]Since \(f\left(-\frac{1}{2}\right) = -1\), this option does not satisfy the condition.

Step 4: Evaluate Option D

Substitute \(x = -\frac{1}{2}\) into the fourth function, \(f(x) = [2x]\).\[ f\left(-\frac{1}{2}\right) = \left\lfloor 2 \times \left(-\frac{1}{2}\right) \right\rfloor = \left\lfloor -1 \right\rfloor = -1 \]Since \(f\left(-\frac{1}{2}\right) = -1\), this option does not satisfy the condition.

Key Concepts

Absolute Value

The concept of absolute value involves the distance of a number from zero on the number line, without considering direction. It's a way to express how far a number is from zero, so it is always non-negative. When evaluating the absolute value of a product like in the problem, such as \(-2x\), it's important to remember:

• If the number inside the absolute value is positive, the absolute value is simply the same number.
• If the number is negative, the absolute value is the positive counterpart of the number.

For instance, in the step by step solution, we found \(|-2 \times (-\frac{1}{2})| = |1| = 1\). This shows how the absolute value "flips" the negative result to a positive form, leading to the final result of 1, which is different from -1 as stated in the question.

Floor Function

The floor function, denoted by \([x]\), is a mathematical operation that rounds a number down to the nearest integer. It's particularly useful when you need the largest integer not greater than a given number.Here are some key points about the floor function:

• It always rounds towards zero, meaning that it "drops" the decimal part and takes only the whole number part.
• If dealing with a negative number, the result is the next integer less than the number itself.

For example, in the exercise you might remember that \(\left\lfloor -\frac{1}{2} \right\rfloor = -1\). This is because -1 is the greatest integer less than or equal to -0.5. Comprehending the floor function is fundamental to evaluating expressions that round down numbers efficiently.

Substitution in Functions

Substitution in functions is a straightforward yet critical process, where we take an input value and plug it into a function to evaluate it. This is akin to following a recipe. By substituting different values into the function's formula, we can easily determine the function's output.Consider how substitution simplifies problem-solving:

• Replace the variable in the function, often denoted as \(x\), with a given number.
• Simplify the expression step by step to find the result.

In the exercise step-by-step solution, substitution involved entering \(x = -\frac{1}{2}\) into four different functions instead of manually solving each function repeatedly. This consistency ensures accurate results.

Function Comparison

Function comparison involves examining two functions to see how their outputs differ when given the same input. It's like a way of measuring and contrasting different responses from multiple rules or formulas.This exercise illustrates function comparison by evaluating four functions:

• Plugging in the same input \(-\frac{1}{2}\) into each function.
• Observing how each function processes this input differently to give distinct results.

Option B demonstrated its uniqueness by yielding a result, \(|1|\), that did not match the expected \(-1\). Function comparison enabled this discrepancy to be highlighted, revealing which function did not satisfy the given condition of equality in this exercise. By running these evaluations, different behavior characteristics of each function become apparent, leading to a reasoned conclusion.