Functions can be expressed using a special notation called function notation, which provides a clear way to represent the relationship between inputs and outputs. Instead of writing an equation as y = 3x - 1, we use the notation f(x) = 3x - 1, where f is the name of the function and x is the input variable. This notation is read as "f of x" and represents the output value corresponding to the input x.

To evaluate a function at a specific input, such as f(4), you substitute the input value into the function's expression. For example, given f(x) = 3x - 1, substituting x = 4 yields:

$$f(4) = 3 \times 4 - 1 = 12 - 1 = 11$$

This means when the input is 4, the output of the function is 11. Function notation is versatile and can use different letters to name functions, such as g(x) or h(x), but the process of substitution and evaluation remains the same.

Functions can also be represented as sets of ordered pairs, where each pair consists of an input (x-value) and its corresponding output (y-value). For example, if a function g is given by the set of ordered pairs, to find g(1), you locate the pair where the input is 1 and identify the output. If the pair is (1, 5), then g(1) = 5.

When functions are graphed, the input corresponds to the x-coordinate, and the output corresponds to the y-coordinate on the graph. To find h(-2) for a function h(x) graphed as a curve, locate the point where x = -2 on the x-axis, then find the corresponding y-value on the curve. If the y-value is 3, then h(-2) = 3.

Understanding function notation is essential for working with functions in algebra and calculus. It allows you to clearly express and evaluate functions, whether given by equations, ordered pairs, or graphs. Remember, the key idea is that the function takes an input x and produces an output, often denoted as f(x), g(x), or h(x), depending on the function's name.

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