91Ó°ÊÓ

When studying the domain of a function, we're identifying all the x-values for which the function is defined. For the given function, \( h(x) = \frac{1}{1-e^{x}} \), the denominator must not be zero as this would make the function undefined. Therefore, we set \( 1 - e^{x} eq 0 \). Solving \( 1 - e^x = 0 \) gives us \( e^x = 1 \) or \( x = 0 \). This means the function is defined for all real numbers except \( x = 0 \). Thus, the domain is:

• All real numbers \( \mathbb{R} \) except \( x = 0 \).

The range of a function tells us all possible outputs or y-values. By looking at the behavior of \( h(x) \), as \( x \to -\infty \), \( h(x) \to -1 \). As \( x \to 0^{-} \), it goes towards \( \infty \), and as \( x \to 0^{+} \), it heads to \( -\infty \). Therefore, the range is:

• All real numbers except \( h(x) = -1 \).

Intercepts are points where the graph intersects the x-axis and y-axis. To find these, we can set the function equal to zero for x-intercepts and set \( x = 0 \) for y-intercepts. Let's talk about each:

• X-intercepts: These occur where \( h(x) = 0 \). For \( \frac{1}{1-e^{x}} = 0 \), it would imply an undefined situation as a fraction equals zero when its numerator is zero, and here the numerator doesn't contain x. Hence, there are no x-intercepts for this function.
• Y-intercepts: Setting \( x = 0 \) discovers \( h(0) \), however \( h(x) \) is undefined at \( x = 0 \). Thus, there are no y-intercepts in this case.

Graph sketching relies heavily on understanding asymptotic behavior, domain, range, and intercepts. Here’s how we can construct a basic sketch for \( h(x) = \frac{1}{1-e^{x}} \):

• Vertical Asymptote: At \( x = 0 \), the function has a vertical asymptote, since the denominator becomes zero. This means the graph will approach this line but never touch it.
• Horizontal Asymptote: As \( x \to \infty \), the value of \( e^{x} \) grows quickly, making the fraction approach a y-value of \( -1 \). Thus, \( y = -1 \) is a horizontal asymptote.
With these asymptotes, the graph should:
• Approach \( -1 \) as \( x \to \infty \).
• Move toward \( +\infty \) as \( x \to 0^{-} \) and toward \( -\infty \) as \( x \to 0^{+} \).
Start by plotting the asymptotes and sketch the curve to accurately reflect how it behaves near these lines.