91Ó°ÊÓ

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In the function \( y=\left(\frac{1}{3}\right)^{x} \), the base \( \frac{1}{3} \) is less than 1, indicating that this is a case of exponential decay. Exponential decay occurs when the value of \( y \) decreases as \( x \) increases. Specifically, for \( x = 0 \), the function evaluates to 1, because any non-zero number raised to the power of zero equals one. This forms the point (0, 1) on the graph.

• The greater the base, the slower the decay; here the base \( \frac{1}{3} \) means the graph drops off fairly quickly as \( x \) increases.
• Exponential functions are daily observed in real life through processes like radioactive decay and depreciation of economies.

When graphing exponential functions, one should note they never touch the x-axis, hence they don't have an x-intercept. They asymptotically approach the x-axis as the exponent increases.

Logarithmic functions are the inverses of exponential functions. The function \( y=\log_{1/3} x \) involves finding the exponent of the base \( \frac{1}{3} \) that gives you \( x \). This function is defined only for positive values of \( x \), indicating that the domain is \( x > 0 \). It effectively 'undoes' the operation of its exponential counterpart. The point (1,0) is an important feature here because the logarithm of 1 is always 0, aligning with the inverse relationship.

• When graphed, logarithmic functions will pass through (1,0) and rise steeply as \( x \) approaches 0 from the right.
• They are often used in fields like acoustics for measuring sound intensity or in seismology for earthquake magnitudes.

Remember, a logarithmic function has no y-intercept since it becomes undefined for negative or zero values of \( x \).

Graphing functions involves plotting mathematical expressions on a coordinate plane to visualize their behavior. When dealing with inverse functions like our exponential \( y=\left(\frac{1}{3}\right)^{x} \) and logarithmic \( y=\log_{1/3} x \), symmetry plays a pivotal role. Inverse functions reflect over the line \( y=x \), meaning both functions should mirror each other across this diagonal line.

• Exponential graph should curve downward while the logarithmic graph should rise steeply.
• For exponential functions, focus on the rapid decrease as \( x \) increases.
• For logarithmic graphs, the focus is on the steep rise as \( x \) approaches zero from the positive side.

This symmetry check helps ensure your graphs are accurate. By visualizing these functions on the same axes, you can more easily understand their inverse relationship and the natural behavior of these mathematical models.