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Understanding what a linear function is serves as the foundation for evaluating its values. Imagine drawing a straight line on a graph; this is essentially what a linear function is. It is an algebraic equation where each term is either a constant or the product of a constant and a single variable.

Mathematically, a linear function is often written in the form of f(x) = mx + b, where m is the slope and b is the y-intercept. The slope m indicates how steep the line is, and the y-intercept b tells us where the line crosses the y-axis.

In the provided exercise, f(x) = 3x - 10, the slope is 3, which means the line goes up by 3 units on the y-axis for every unit it moves along the x-axis. The y-intercept is -10, placing the point where the line crosses the y-axis 10 units below the origin.

Function substitution is a critical skill when working with functions. It's the process of replacing the variable in the function with a given number or expression. To substitute a value into a function, you simply replace every occurrence of the variable with that value.

Why is this Useful?

Function substitution allows us to find the output of a function for any particular input. This is especially useful in situations where you might not have a graph in front of you or when dealing with complex functions where drawing a graph isn't feasible.

For example, if you're given f(x) = 3x - 10 and you want to evaluate it for x = 0, you'll replace every x in the equation with 0, transforming the equation into f(0) = 3(0) - 10. This makes it clear what the next step is: calculation.

Once we’ve substituted the values into the function, we then evaluate, or simplify, it to find the resulting value. It's like following a recipe: once you've got your ingredients substituted in, it's time to cook and see the final dish.

Steps in Function Evaluation

Always perform the operations in the correct order. Begin with any multiplication or division from left to right, and then move on to addition or subtraction, following the order of operations. In our example with f(0) = 3(0) - 10, begin by multiplying 3 by 0, which equals 0, and then subtract 10 to end up with -10.

This process of evaluating the function gives us the output value for our given input, allowing us to understand how the function behaves at any particular point. Continuing with our linear function example, when x = 0, the function's value is -10, which also corresponds to the y-intercept of the graph of the function, solidifying our understanding of the relationship between algebraic evaluation and graphical representation.