Find the domain and sketch the graph of the function. f(x)= 3-(1)/(2) x if x ⩽2 2 x-5

step 1 For the following problems, we're looking to find the domain of the graph. And oftentimes the do

Find the domain and sketch the graph of the function. f(x)=...

Key Concepts

Domain

The domain of a function represents the set of all input values (typically x-values) for which the function is defined. In many cases, including functions defined by different rules over different intervals, the domain is determined by considering all the pieces and ensuring that every input value falls into one of the pieces' conditions.

Piecewise-Defined Function

A piecewise-defined function is one that is described by different expressions depending on the input interval. This type of function requires careful consideration of each rule and the specific condition or interval applied, as each piece may behave differently. Understanding piecewise functions is fundamental for analyzing and sketching functions that have different behaviors in different parts of their domain.

Linear Functions

Linear functions are basic functions of the form f(x) = mx + b, where m is the slope and b is the y-intercept. In the context of piecewise-defined functions, each linear segment exhibits a straight-line behavior, making them easier to graph and analyze. Recognizing the properties of linear functions helps in understanding the overall behavior of the function on different intervals.

Graphing Piecewise Functions

Graphing piecewise functions involves plotting each segment of the function over its designated interval and carefully indicating whether the endpoints are included or excluded. Special attention must be given to the boundaries where the function's rule changes, ensuring that the correct open or closed circles are used to represent the inclusion or exclusion of points.

Continuity and Endpoint Behavior

When dealing with piecewise functions, it is important to examine the behavior of the function at the boundaries between different pieces. This includes checking if the function is continuous (if the segments meet without a gap) or if there is a jump discontinuity. Assessing endpoint behavior helps in accurately sketching the graph and understanding how the function behaves at the transition points.

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