Question

drag and drop the correct words or expressions to complete the statement.

• the y - intercept of the exponential function is $(0,3)$
• as $x\to -\infty$,
• as $x\to \infty$,

$f(x)\to 0$ $(0,5)$ $f(x)\to 4$ $f(x)\to -\infty$ $f(x)\to \infty$

Explanation:

Step1: Analyze the behavior as \(x\to-\infty\)

For an exponential function \(y = a\cdot b^{x}+c\) (\(b> 1\)), as \(x\to-\infty\), \(b^{x}\to0\). If the horizontal asymptote is \(y = 0\) (assuming a simple form \(y=a\cdot b^{x}\) with \(a = 3\) and \(b>1\)), as \(x\to-\infty\), \(f(x)\to0\)

Step2: Analyze the behavior as \(x\to\infty\)

For an exponential function \(y=a\cdot b^{x}\) (\(a>0,b > 1\)), as \(x\to\infty\), \(b^{x}\to\infty\). So \(y=a\cdot b^{x}\to\infty\) (since \(a = 3>0\) and if \(b>1\))

Answer:

As \(x\to-\infty\), \(f(x)\to0\); As \(x\to\infty\), \(f(x)\to\infty\)