Question

Rewrite the piecewise function as an absolute value function.

a) |x + 3|
b) |x - 3|
c) |x| + 3
d) x + |3|

Answer 1

Rewriting a piecewise function as an absolute value function requires identifying the structure of the piecewise function. In this case, all the options given are already in absolute value form or can be expressed simply. For example, the function x + |3| is equivalently written as x + 3. Option D is the correct answer.

Step-by-step explanation:

The task of rewriting a piecewise function as an absolute value function refers to expressing the function using absolute value notation, which captures both the positive and negative branches of a piecewise function. Taking into account the characteristics of the absolute value function, we notice that options |x + 3|, |x - 3|, and |x| + 3 are already in the form of an absolute value. The option x + |3| can also be written as an absolute value function since |3| is a constant and equals 3; therefore, the rewritten absolute value function is simply x + 3.

To answer the separate queries provided:

1. For instance, if we have a function with positive value and decreasing positive slope at x = 3, option (a) y = 13x reflects a constant positive slope, not a decreasing one. Option (b) y = x² has a positive slope that increases with x, so it does not suit the criteria either.
2. The velocity of an object whose position as a function of time is given by x(t) = −3t² m would be the derivative of x(t) with respect to time t, which gives v(t) = dx(t)/dt = −6t m/s. The velocity is never positive because the derivative is negative for all non-zero t. At t = 1.0 s, the velocity is v(1.0) = −6 m/s, and the speed, being the absolute value of velocity, is 6 m/s.

Considering the understanding of translations of functions, none of the sample functions provided directly correspond to rewriting a piecewise function as an absolute value function but offer insights into horizontal translations of functions.

In conclusion, when rewriting a piecewise function as an absolute value function, it's essential to identify the structure of the piecewise function and then apply the properties of absolute values to construct an equivalent expression.