Question

1. consider the curve $y = 5(2^x)$. 13d what is the range of the function? give your answer as an inequality. 13e as $x$ approaches positive infinity, what value does $y$ approach? options: 5, $\infty$, 0, $-\infty$

Explanation:

Step1: Analyze the function \( y = 5(2^x) \)

The function is an exponential function with base \( 2 \), which is greater than \( 1 \). For exponential functions of the form \( a(b^x) \) where \( a>0 \) and \( b > 1 \), as \( x \) increases (approaches positive infinity), \( b^x \) grows without bound.

Step2: Determine the limit as \( x \to +\infty \)

For \( y = 5(2^x) \), when \( x \) approaches positive infinity, \( 2^x \) approaches \( +\infty \). Multiplying by \( 5 \) (a positive constant) still gives a value that approaches \( +\infty \). So \( y \) approaches \( \infty \).

Answer:

\(\infty\) (the option with \(\infty\))