Question

Sketch the graph of the function. $$ g(x)=-e^{x / 2} $$

Short answer

The graph starts at (0, -1) and decreases rapidly as x increases, slowly approaching 0 as x decreases.

Step-by-step solution

Identify behavior at x = 0

Evaluate the function at x = 0, i.e., when \( x=0, g(x) = -e^{0 / 2} = -1 \). This point will be part of our graph, allowing us to start identifying the function's behavior.

Identify behavior as x increases

As x moves towards positive infinity, the value \( g(x) = -e^{x / 2} \) will decrease and approach negative infinity. The reason for this is the exponential part \( e^{x/2} \) which grows, but then it's negated by the '-' sign, resulting in a decrease in value.

Identify behavior as x decreases

As x moves towards negative infinity, the value \( g(x) = -e^{x / 2} \) will approach 0. The exponential \( e^{x/2} \) becomes smaller as x decreases, but the '-' sign keeps the value negative, so it approaches zero from below.

Sketch the curve

Knowing these trends, we can sketch the function starting at (0, -1), decreasing rapidly as x increases, and slowly approaching 0 as x decreases.