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A function in math describes a special relationship between an input (often called \( x \)) and an output (\( g(x) \)). Each input is related to exactly one output. Functions help us map one value to another, and they follow certain rules or equations.
For example, in the function \( g(x) = 3x - 14 \), every value of \( x \) you choose will produce a specific value by multiplying \( x \) by 3 and then subtracting 14.
This function is linear, meaning it creates a straight line when graphed, and follows the structure of \( ax + b \), where \( a \) and \( b \) are constants.

Solving an equation involves finding the value(s) of \( x \) that make the equation true. Here, we're looking for a fixed point, which is where the function's output equals the input.
For our exercise, we set up the equation \( g(x) = x \), which becomes \( 3x - 14 = x \). Solving involves rearranging the equation to isolate \( x \) on one side. This helps us find the exact point where our input matches the output.

• First, we subtract \( x \) from both sides: \( 2x - 14 = 0 \).
• Next, add 14 to make \( 2x = 14 \).
• Finally, divide by 2 to find \( x = 7 \).

Here, \( x = 7 \) satisfies the equality, making it the fixed point of the function.

Real numbers are values that represent a continuous line on the number line. They include positive and negative numbers, whole numbers, and fractions, essentially any number that is not imaginary.
In our exercise, we're asked to find fixed points that are real numbers. This means that the solutions we find should fall into this set of numbers.
When solving the function, we found \( x = 7 \). This falls within the category of real numbers, making it a valid solution. Real numbers are a key concept in understanding the types of solutions we can expect in math problems like this one.
This concept reassures us that every solution we're considering must end up being a true real world value, which is important because it ensures our equations have practical and interpretable solutions.