91Ó°ÊÓ

The domain of an exponential function like \( f(x) = -2^x \) is all real numbers. This means you can pick any real number for \( x \), and there will always be a corresponding \( f(x) \). In other words, you can select any number along the x-axis, whether it's negative, zero, or positive, and plug it into the function.

For the range, consider the nature of the function \( f(x) = -2^x \). Normally, \( 2^x \) is always positive because raising a positive number to any power never results in a negative answer. However, when you multiply it by \( -1 \), every value of \( f(x) \) becomes negative.

This means that the outputs, or \( f(x) \) values, are always negative real numbers, and the range is \( (-\infty, 0) \). Overall, the domain encompasses every input value on the x-axis, while the range covers every output value on the negative y-axis.

A key feature of exponential functions is the horizontal asymptote, a value that the function gets infinitely close to but never truly reaches. For \( f(x) = -2^x \), the asymptote is at \( y = 0 \).

As you evaluate what happens when \( x \) moves towards positive or negative infinity, remember that \( 2^x \) grows rapidly larger as \( x \) increases and quickly approaches zero as \( x \) becomes very negative.

Multiplying \( 2^x \) by \( -1 \) means \( -2^x \) will appear below the x-axis, approaching zero from the negative side. Although the function gets closer and closer to \( y = 0 \), it will not cross the x-axis, remaining asymptotic to it.

When sketching the graph of \( f(x) = -2^x \), start by plotting some key points to guide you. Consider easy values for \( x \), such as \( x = 0 \), \( x = 1 \), and \( x = -1 \). Calculate their corresponding \( f(x) \):

• At \( x = 0 \), \( f(0) = -1 \).
• At \( x = 1 \), \( f(1) = -2 \).
• At \( x = -1 \), \( f(-1) = -0.5 \).

With these points plotted, think about the function's properties. The positive exponential \( 2^x \) curves upwards, but since it's multiplied by \( -1 \), \( -2^x \) reflects across the x-axis. This causes the curve to descend as \( x \) becomes more positive.

The curve should smoothly approach the horizontal asymptote at \( y = 0 \) as \( x \) goes to infinity. Make sure the line suggests a continuing approach toward \( y = 0 \), getting closer but never touching or crossing. This provides the graph its characteristic exponentially decaying shape.