Question

In Exercises 45-48, find (a) a simple basic function as a right end behavior model and (b) a simple basic function as a left end behavior model for the function. $$y=e^{x}-2 x$$

Short answer

The right end behavior model for the function \(y=e^{x}-2x\) is \(e^{x}\), and the left end behavior model is \(-2x\).

Step-by-step solution

Right End Behavior Model

Consider the function as \(x\) approaches positive infinity. As \(x\) increases without bound, the term \(e^{x}\) will dominate the term \(-2x\). So, at large \(x\) values, the function behaves like \(e^{x}\). Hence, \(e^{x}\) is a simple basic function that describes the right end behavior of the function.

Left End Behavior Model

Consider the function as \(x\) approaches negative infinity. As \(x\) decreases without bound, the term \(e^{x}\) will tend towards 0, and the \(-2x\) term will dominate. Hence at large negative \(x\) values, the function behaves like \(-2x\). So, \(-2x\) is a simple basic function that describes the left end behavior of the function.