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When discussing functions, understanding the domain and range is key. The **domain** of a function is the set of all possible input values (usually x-values) that the function can accept. For the linear function given by \(f(x) = 2x - 3\), there are no restrictions on the input, meaning the domain is all real numbers. In interval notation, this is represented as

• Domain of \(f(x)\): \((-\infty, \infty)\).

The **range** of a function is the set of all possible output values (usually y-values). Since the function given is linear and continues indefinitely in both directions, the y-values also span all real numbers. Thus, the range of \(f(x)\) is also

• Range of \(f(x)\): \((-\infty, \infty)\).

The inverse function \(f^{-1}(x) = \frac{x+3}{2}\) similarly has a domain and range of all real numbers:

• Domain of \(f^{-1}(x)\): \((-\infty, \infty)\)
• Range of \(f^{-1}(x)\): \((-\infty, \infty)\)

Visualizing a function can significantly help in understanding its behavior, and graphing is a useful tool for this. For the linear function \(f(x) = 2x - 3\), the graph is a straight line with a slope of 2 and a y-intercept at -3. Graphs of linear functions are particularly simple as they maintain a constant slope throughout.

To graph, you can plot points by substituting values of x into the function. For instance, when \(x = 0\), \(f(x) = -3\), which is your y-intercept, and when \(x = 1\), \(f(x) = -1\). Drawing a line through these points will represent the function.
Similarly, for the inverse function \(f^{-1}(x) = \frac{x + 3}{2}\), the slope is \(\frac{1}{2}\) and the y-intercept is \(\frac{3}{2}\). Plot some points for this function and draw the line. These two lines will be mirror reflections across the line \(y=x\), which is a helpful visual check when ensuring the functions are correctly graphed.

Graphing these functions together helps visualize the relationship between a function and its inverse.

Linear functions form the foundation of algebraic functions, defined by the general form \(f(x) = mx + c\), where \(m\) is the slope, and \(c\) is the y-intercept. They are characterized by constant rates of change and produce straight-line graphs.

• **Slope (m):** Indicates the steepness and direction of the line. Positive slopes incline upward to the right, and negative slopes incline downward to the right.


• **Y-intercept (c):** The point where the line crosses the y-axis. For the function \(f(x) = 2x - 3\), the y-intercept is -3.

By analyzing the slope and intercept, you can quickly sketch and understand the behavior of linear functions. In our specific problem, the slope of 2 means that for every 1 unit increase in \(x\), \(y\) increases by 2 units. This provides a clear, step-by-step approach to plotting and interpreting linear equations.