In Exercises 1–30, find the domain of each function. f(x)=3(x-4)

Verified step by step guidance

1. Identify the type of function given. The function \( f(x) = 3(x-4) \) is a linear function.
2. Recall that the domain of a linear function is all real numbers because there are no restrictions on the values that \( x \) can take.
3. Consider any potential restrictions. For linear functions, there are no denominators or square roots that could restrict the domain.
4. Conclude that since there are no restrictions, the domain of \( f(x) = 3(x-4) \) is all real numbers.
5. Express the domain in interval notation: \((-\infty, \infty)\).

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For polynomial functions like f(x) = 3(x - 4), the domain typically includes all real numbers, as there are no restrictions such as division by zero or square roots of negative numbers.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The function f(x) = 3(x - 4) is a linear polynomial, which is a specific type of polynomial of degree one. Polynomial functions are continuous and defined for all real numbers.

Graphing Linear Functions

Graphing linear functions involves plotting points that satisfy the function's equation and connecting them to form a straight line. The function f(x) = 3(x - 4) can be rewritten in slope-intercept form as f(x) = 3x - 12, indicating a slope of 3 and a y-intercept of -12. Understanding the graph helps visualize the domain and behavior of the function.