Question

Multiple Choice Which of the following gives the range of \(y=4-2^{-x} ?\) \((\mathbf{A})(-\infty, \infty) \quad(\mathbf{B})(-\infty, 4) \quad(\mathbf{C})[-4, \infty)\) \((\mathbf{D})(-\infty, 4]\) (E) all reals

Short answer

The correct answer is choice D

Step-by-step solution

Identify the Function's Behavior for Large Positive x-Values

As x approaches to positive infinity, the term \(2^{-x}\) will approach zero because any non-zero number raised to an increasingly large negative exponent approaches zero. Therefore, \(4 - 2^{-x}\) will get closer and closer to 4.

Identify the Function's Behavior for Large Negative x-Values

As x approaches to negative infinity, the term \(2^{-x}\) will approach to positive infinity because any number (except 0) raised to an increasingly large positive exponent approaches infinity. But \(4 - 2^{-x}\) will get closer and closer to negative infinity.

List the Possible y-Values (The Range)

Combining the information from steps 1 and 2, as x ranges over all real numbers, the possible y-values will range from negative infinity up to (and including) 4. This defines the range of the function and it corresponds to choice \(D)(-\infty, 4].\)