Question

The function \(f\) is given by \(f(x)=2-|x-4| .\) For what value of \(x\) does the function \(f\) achieve its maximum value? (A) 2 (B) 4 (C) 5 (D) 6

Short answer

The function achieves its maximum value at \(x = 4\).

Step-by-step solution

Understanding the Function

The function given is \(f(x) = 2 - |x - 4|\). It includes an absolute value term \(|x - 4|\), which means it contains a linear function both above and below the point \(x = 4\). The piecewise nature of \(|x-4|\) results in a V-shaped graph.

Determine Break Point of the Absolute Value

The expression \(|x - 4|\) changes behavior at \(x = 4\). For \(x \geq 4\), \(|x-4| = x - 4\), and for \(x < 4\), \(|x-4| = -(x-4)\). This makes \(x = 4\) the critical value where the behaviour of the graph changes.

Evaluate Function at Critical Point

Calculate \(f(x)\) at \(x = 4\) to find if it is a maximum. Substituting into the function, \(f(4) = 2 - |4 - 4| = 2 - 0 = 2\).

Check Behavior Around the Critical Point

At \(x < 4\), \(|x-4| = 4-x\), thus the function becomes \(f(x) = 2 - (4 - x) = x - 2\), increasing as \(x o 4^{-}\). At \(x > 4\), \(|x-4| = x - 4\) and the function is \(f(x) = 2 - (x-4) = 6 - x\), decreasing as \(x o 4^{+}\).

Conclusion About Maximum

Since the function is increasing on the left side of \(x=4\) and decreasing on the right side, it reaches its maximum value at \(x=4\). Thus, the maximum value is 2, which occurs at \(x = 4\).

Key concepts

Absolute Value Functions

Absolute value functions can be a bit tricky because they involve an absolute value sign, which affects the function's behavior. The absolute value of a number is its distance from zero on the number line, which means it is always non-negative. When you see a function like \[ f(x) = 2 - |x-4| \] it indicates that the distance of \( x \) from 4 is being subtracted from 2. This specific function is a classic example of how absolute value functions create a V-shaped graph. The point at the bottom of the "V" is the vertex.

• The absolute value function \(|x - 4|\) evaluates to \(x-4\) when \(x \geq 4\), while it becomes \(-(x-4)\) for \(x < 4\).
• This behavior switch at \(x = 4\) is what makes it a piecewise function.

Even though absolute values always output non-negative results, the function itself can have negative values depending on the operations surrounding the absolute value.

Piecewise Functions

Piecewise functions are functions that are defined by different equations over different intervals. These equations change according to the value of \(x\). This kind of function is very useful in representing situations where a rule or behavior changes at a certain point. For the function \(f(x) = 2 - |x-4|\), it can be split into two pieces depending on the value of \(x\).

• When \(x < 4\), the expression inside the absolute value becomes \(|x-4| = -(x-4) = 4-x\), so the function becomes \(f(x) = x-2\). Here, as \(x\) lowers to 4, \(f(x)\) pulls upwards.
• When \(x \geq 4\), the function behaves like \(|x - 4| = x - 4\), making \(f(x) = 6 - x\). As \(x\) increases past 4, \(f(x)\) pushes downwards.

By recognizing the piecewise nature of functions, you can break them down into simple linear equations which describe their behavior over distinct intervals.

Function Maximum

Finding the maximum value of a function involves determining the highest point it reaches on its graph. For a piecewise function with absolute values like \(f(x) = 2 - |x-4|\), understanding where its graph shifts behavior is key to identifying this maximum.

• The graph of \(f(x)\) is divided by \(x = 4\).At this dividing point, the absolute value components create clear upward and downward trends.
• The direction of these trends indicate that as \(x\) approaches 4 from the left, the function value increases; and from the right, it decreases.
• A behavior like this, where the function value increases until a certain point and then decreases, suggests a maximum at the changeover point.

Thus, the maximum is located at \(x=4\), where \(f(x) = 2\). Understanding function maximums can greatly simplify problems involving function graphs and their crucial features.