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Let's explore constant functions like the one in the given equation: \( g(x) = -4 \). A constant function is a type of linear function where the output value (or y-value) remains the same, no matter what x-value is input.
In this case, the function always outputs -4. This means that for any value of x, the y-value will always be -4. You can visualize this with a horizontal line on a graph. It's horizontal because the y-value does not change. For example:

• For \( x = 0 \), \( g(x) = -4 \)
• For \( x = 5 \), \( g(x) = -4 \)
• For \( x = -3 \), \( g(x) = -4 \)

No matter what x you choose, the y-value remains constant. This is the defining feature of constant functions. They are represented graphically by horizontal lines and are straightforward to understand and graph.

Understanding the domain and range of a function is crucial.
Domain: The domain of a function is the set of all possible input values (x-values). For the function \( g(x) = -4 \), there are no restrictions on the x-values. You can put any real number into the function, and it will give you an output of -4. Therefore, the domain of \( g(x) = -4 \) is all real numbers. In interval notation, we write this as \( (-fty, fty) \).
Range: The range of a function is the set of possible output values (y-values). Since \( g(x) = -4 \) only outputs the value -4 regardless of what x-value is chosen, the range is simply \{ -4 \}. This tells us that no matter what x is, the y-value of the function will always be -4.
Knowing the domain and range helps us understand the behavior and limitations of the function.

To graph functions, we use the coordinate plane. The coordinate plane is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Here’s how to use it for our function, \( g(x) = -4 \):

• Start by drawing the x-axis and y-axis. These axes intersect at the origin \( (0,0) \).
• Next, identify the y-value of the function, which is -4 in this case.
• Draw a horizontal line that passes through y = -4. This line should be parallel to the x-axis.

The coordinate plane allows us to visualize relationships between x-values and y-values. For example:

• Point (1, -4) lies along this line, with x = 1 and y = -4.
• Point (2, -4) also lies on this line, with x = 2 and y = -4.

By plotting these points, we create a graphical representation of the function. The coordinate plane helps make the behavior of the function more concrete and understandable.