Question

1. consider the curve $y = 5(2^x)$. 13d what is the range of the function? give your answer as an inequality.

Explanation:

Step1: Analyze the exponential function \(2^x\)

The exponential function \(y = 2^x\) has a range of \(y>0\) for all real \(x\) because any positive number raised to a real power is positive, and as \(x\) approaches \(-\infty\), \(2^x\) approaches \(0\) (but never reaches \(0\)), and as \(x\) increases, \(2^x\) increases without bound.

Step2: Analyze the function \(y = 5(2^x)\)

We multiply the function \(2^x\) by \(5\). Since multiplying a positive number (because \(2^x>0\)) by \(5\) (a positive constant) will still result in a positive number, and as \(x\) approaches \(-\infty\), \(5\times2^x\) approaches \(0\) (but never reaches \(0\)), and as \(x\) increases, \(5\times2^x\) increases without bound. So the range of \(y = 5(2^x)\) is \(y>0\).

Answer:

\(y > 0\)