Questions: f(x) = x+3 + x-3 + x^2 - 1

Solution Steps

To solve the problem of evaluating the function \( f(x) = |x+3| + |x-3| + |x^2-1| \), we need to consider the behavior of the absolute value expressions. The function is piecewise, and its behavior changes at the points where the expressions inside the absolute values are zero. These points are \( x = -3 \), \( x = 3 \), and \( x = \pm 1 \). We will evaluate the function in the intervals determined by these critical points.

We evaluate the function \( f(x) = |x+3| + |x-3| + |x^2-1| \) at the critical points and selected values:

• For \( x = -4 \): \[ f(-4) = |-4 + 3| + |-4 - 3| + |(-4)^2 - 1| = | -1 | + | -7 | + |16 - 1| = 1 + 7 + 15 = 23 \]
• For \( x = -3 \): \[ f(-3) = |-3 + 3| + |-3 - 3| + |(-3)^2 - 1| = |0| + |-6| + |9 - 1| = 0 + 6 + 8 = 14 \]
• For \( x = -2 \): \[ f(-2) = |-2 + 3| + |-2 - 3| + |(-2)^2 - 1| = |1| + |-5| + |4 - 1| = 1 + 5 + 3 = 9 \]
• For \( x = 0 \): \[ f(0) = |0 + 3| + |0 - 3| + |0^2 - 1| = |3| + |-3| + |-1| = 3 + 3 + 1 = 7 \]
• For \( x = 1 \): \[ f(1) = |1 + 3| + |1 - 3| + |1^2 - 1| = |4| + |-2| + |0| = 4 + 2 + 0 = 6 \]
• For \( x = 2 \): \[ f(2) = |2 + 3| + |2 - 3| + |2^2 - 1| = |5| + |-1| + |4 - 1| = 5 + 1 + 3 = 9 \]
• For \( x = 3 \): \[ f(3) = |3 + 3| + |3 - 3| + |3^2 - 1| = |6| + |0| + |9 - 1| = 6 + 0 + 8 = 14 \]
• For \( x = 4 \): \[ f(4) = |4 + 3| + |4 - 3| + |4^2 - 1| = |7| + |1| + |16 - 1| = 7 + 1 + 15 = 23 \]

The results of the evaluations are:

• \( f(-4) = 23 \)
• \( f(-3) = 14 \)
• \( f(-2) = 9 \)
• \( f(0) = 7 \)
• \( f(1) = 6 \)
• \( f(2) = 9 \)
• \( f(3) = 14 \)
• \( f(4) = 23 \)

Final Answer