Questions: For the function f(x)=x^2-4x+4, complete parts (a)-(d) below.

Transcript text: For the function $f(x)=\left|x^{2}-4 x+4\right|$, complete parts (a)-(d) below.

Solution

Solution Steps

To solve the given function $ f(x) = |x^2 - 4x + 4| $, we need to analyze the expression inside the absolute value. The expression $ x^2 - 4x + 4 $ is a quadratic equation, which can be factored or expanded to understand its behavior. We will find the roots of the quadratic equation to determine where the expression changes sign, and then evaluate the absolute value function accordingly.

Step 1: Analyze the Quadratic Expression

The function given is $ f(x) = |x^2 - 4x + 4| $. The expression inside the absolute value is a quadratic equation: $ x^2 - 4x + 4 $. This can be rewritten as $ (x-2)^2 $, indicating that it is a perfect square.

Step 2: Find the Roots of the Quadratic

The roots of the quadratic equation $ x^2 - 4x + 4 = 0 $ are found by solving $ (x-2)^2 = 0 $. This gives a double root at $ x = 2 $.

Step 3: Evaluate the Absolute Value Function

Since $ (x-2)^2 $ is always non-negative, the absolute value function $ f(x) = |(x-2)^2| $ simplifies to $ f(x) = (x-2)^2 $. Therefore, the function is non-negative for all $ x $.

Step 4: Determine the Behavior of the Function

The function $ f(x) = (x-2)^2 $ reaches its minimum value at the root $ x = 2 $, where $ f(2) = 0 $. For all other values of $ x $, $ f(x) > 0 $.