Problem 35 - Algebra A Combined Function

Question

Graph each piecewise-defined function. Use the graph to determine the domain and range of the function. $$ f(x)=\left\\{\begin{array}{rll} -2 x & \text { if } & x \leq 0 \\ 2 x+1 & \text { if } & x>0 \end{array}\right. $$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\); Range: \((-\infty, 0] \cup (1, \infty)\).

Step 1: Understand the Function

The given function is piecewise-defined, meaning it is defined by different expressions based on the value of the input variable, \(x\). It has two parts: \(-2x\) when \(x\leq 0\), and \(2x + 1\) when \(x > 0\).

Step 2: Graph the First Piece

For the first piece, \(f(x) = -2x\) when \(x \leq 0\). Plot points by substituting values for \(x\) such as \(-2\), \(-1\), and \(0\): \((-2, 4), (-1, 2), (0, 0)\). Draw a ray to the left starting from the point \((0, 0)\) with a closed dot at the endpoint, since \(x = 0\) is included.

Step 3: Graph the Second Piece

For the second piece, \(f(x) = 2x + 1\) when \(x > 0\). Choose positive values for \(x\) like \(1\) and \(2\): \((1, 3), (2, 5)\). Draw a ray to the right starting from an open circle at \((0, 1)\) since \(x = 0\) is not included.

Step 4: Determine the Domain

The domain of the function includes all real numbers because there are no restrictions on \(x\) from the pieces; one covers \(x \leq 0\) and the other \(x > 0\). Thus, the domain is all real numbers, \((-\infty, \infty)\).

Step 5: Determine the Range

The range of the function can be identified from the graph. The line for \(x \leq 0\) results in \(y\) values from \(0\) upwards in the negative direction. The line for \(x > 0\) results in \(y\) values starting just above \(1\). Therefore, the range is \((-\infty, 0] \cup (1, \infty)\).

Key Concepts

Domain and Range

To understand the domain and range, we first need to know what these terms mean in mathematics.
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
In this problem, the domain for the piecewise function is all real numbers. This means there are no restrictions on the x-values. The different parts of the function cover all x-values together.The range of a function is the set of all possible output values (y-values) that the function can produce.
For this piecewise function, when we look at each part separately, the range becomes easier to determine from the graph:

• For the part \(f(x) = -2x\) when \(x \leq 0\), \(y\) can be any value from 0 downwards to negative infinity.
• For the part \(f(x) = 2x + 1\) when \(x > 0\), \(y\) starts just above 1 and increases indefinitely.

The combined range is \((-\infty, 0] \cup (1, \infty)\), indicating that every value from negative infinity up to 0 and from just above 1 onward can be achieved.

Graphing Functions

Graphing piecewise functions can initially seem daunting, but understanding each part separately helps.
The graph of each expression within the function is drawn over the specified intervals of x.
Using the given piecewise function as an example:

• For \(f(x) = -2x\) when \(x \leq 0\), plot several points using different x values like \(-2, -1,\) and \(0\). These have corresponding y-values \/((2, 4), (-1, 2), \/ and \(0,0)\). Connect these points to form a ray heading left from \(0,0\) and include the endpoint with a closed dot at \(0,0\).
• For \(f(x) = 2x + 1\) when \(x > 0\), use x-values such as \(1\) and \(2\) for which y-values are \((1,3)\) and \((2,5)\). Connect these to create a ray going right, starting with an open circle at \(0,1\) as \(x=0\) is not included in this interval.

By putting these segments together, you get a continuous graph of the piecewise function.

Piecewise-Defined Function

A piecewise-defined function is essentially a function that has different expressions for different pieces of its domain.
These types of functions allow us to define outputs based on specific intervals of the domain.
Understanding the details of each piece helps in constructing the graph accurately and determining the domain and range.In our example, the function is defined:

• By the expression \(f(x) = -2x\) if \(x \leq 0\).
• By \(f(x) = 2x + 1\) for \(x > 0\).

Each piece is graphed separately, adhering to their respective conditions on x-values.
Look out for conditions like closed or open circles at endpoints, which indicate if the endpoint value is included or not in the piece.
It’s the combination of these segments that provides a complete picture of the function as a whole.