91Ó°ÊÓ

To solve equations involving exponential functions, it's essential to understand the properties and behavior of these functions. The equation given in the exercise is an exponential function:
y = 3^x

Exponential functions have the general form y = a^x, where a is a positive constant. They grow rapidly for positive x and decay towards zero for negative x.

When solving for specific values, we substitute the given x or y values into the equation and simplify:

• For y = 3^x with x = 2, calculate y by substituting x: y = 3^2 = 9. So, the ordered pair is (2, 9).
• For y = 3^x with y = 3, solve for x:
  3 = 3^x → x = 1. Hence, the ordered pair is (1, 3).

It's crucial to use the properties of exponential functions in reversing the exponentiation when needed, such as:
3 = 3^x → x = 1

Also, remember that negative exponents indicate taking the reciprocal, such as 3^{-1} = 1/3.

Ordered pairs are essentially points on a coordinate plane represented by two values: (x, y). The first value in the pair is the x-coordinate, and the second is the y-coordinate.

• The pair (2, 9) tells us that when x is 2, y is 9.
• The pair (1, 3) tells us that when x is 1, y is 3.

To determine a missing value in an ordered pair, you can substitute the known value into the equation:

• For (2, ), substituting x = 2 we get y = 9.
• For (, 3), substituting y = 3 we solve x to get 1.

This process helps us find the correct points that satisfy the given equation.

When completing coordinates, our goal is to find the missing value that makes the pair true for the given equation. Here are the steps:

• Start with the known value of x or y.
• Substitute into the equation y = 3^x.

Here are examples from our exercise:

• For (-1, ), given x = -1, substitute: y = 3^{-1} = 1/3. So, the pair is (-1, 1/3).
• For (, 1/9), given y = 1/9, we solve: 1/9 = 3^x. Recognize that 1/9 is 3^{-2}, so x = -2. So, the pair is (-2, 1/9).

By understanding the properties of exponential functions, solving for missing coordinates becomes a straightforward task.