Question

which function is represented by the graph below?
$f(x)= - 2e^{x}$
$f(x)= - 2\ln(x)$
$f(x)= - \ln(2x)$
$f(x)=e^{-2x}$

Explanation:

Step1: Analyze domain

The graph has a vertical - asymptote near \(x = 0\). The domain of \(y=-2\ln(x)\) and \(y =-\ln(2x)\) is \(x>0\), so these can be excluded as the graph has values for negative \(x\).

Step2: Analyze end - behavior

For \(y = e^{-2x}\), as \(x
ightarrow+\infty\), \(y
ightarrow0\) and as \(x
ightarrow-\infty\), \(y
ightarrow+\infty\). For \(y=-2e^{x}\), as \(x
ightarrow-\infty\), \(y
ightarrow0\) and as \(x
ightarrow+\infty\), \(y
ightarrow-\infty\) which matches the graph.

Answer:

\(f(x)=-2e^{x}\)