Question

Graph each function. Be sure to label three points on the graph. $$\text { If } f(x)=\left\\{\begin{array}{ll}-x^{2} & \text { if } x<0 \\\4 & \text { if } x=0 \text { find: } \\ 3 x-2 & \text { if } x>0\end{array}\right.$$ (a) \(f(-3)\) (b) \(f(0)\) (c) \(f(3)\)

Short answer

f(-3) = -9, f(0) = 4, f(3) = 7.

Step-by-step solution

Evaluate f(-3)

To find the value of the function when \(x = -3\), use the piece of the function for \(x < 0\). That part is \(f(x) = -x^2\). Thus, \(f(-3) = -(-3)^2 = -9\).

Evaluate f(0)

To find the value of the function when \(x = 0\), use the part of the function that applies to \(x = 0\). That is \(f(0) = 4\).

Evaluate f(3)

To find the value of the function when \(x = 3\), use the piece of the function for \(x > 0\). That part is \(f(x) = 3x - 2\). Thus, \(f(3) = 3(3) - 2 = 7\).

Plot the Points

On a graph, plot the points \((-3, -9)\), \((0, 4)\), and \((3, 7)\). Connect them according to their respective function pieces. \(f(x) = -x^2\) for \(x < 0\), a point at \(x = 0\) at \(4\), and \(f(x) = 3x - 2\) for \(x > 0\).

Key concepts

graphing piecewise functions

Piecewise functions are functions that have different expressions depending on the input value (x). They can have different behaviors in different intervals of the domain. Let's look at the given function:

If \( f(x) = \left\{ \begin{array}{ll}-x^{2} & \text{if } x < 0 \ 4 & \text{if } x = 0 \ 3x - 2 & \text{if } x > 0\end{array}\right. \)

When graphing a piecewise function, you need to treat each part independently over its specific interval. Connect the plotted points smoothly according to each respective part of the function. Make sure you follow the rules for each piece:
* For \(x < 0\): Graph the section \(-x^2\). This is a downward-opening parabola. * For \(x = 0\): Plot the specific point \(4\). * For \(x > 0\): Graph the line \(3x - 2\). This is a straight line with slope \(3\) and y-intercept \(-2\).
Remember to label at least three points on your graph to provide accurate and clear visual representation.

evaluating piecewise functions

Evaluating piecewise functions requires you to select the correct piece of the function based on the value of \(x\). Here is how to evaluate the given function step by step:

a) Evaluate \(f(-3)\): Since \(x = -3\) is less than \(0\), use the first piece of the function, \(-x^2\).
\(f(-3) = -(-3)^2 = -9\).
b) Evaluate \(f(0)\): For \(x = 0\), use the second part of the function, which is simply the value \(4\).
So, \(f(0)=4\).
c) Evaluate \(f(3)\): Since \(x = 3\) is greater than \(0\), use the third piece of the function, \(3x - 2\).
Therefore, \(f(3)=3(3)-2=7\).

Each value of \(x\) determines which part of the function you should use. Plug the \(x\) value into the corresponding piece, and solve for \(f(x)\).

plotting points on a graph

Plotting points helps to create an accurate graph of the piecewise function. Here are steps to plot points:

* Take the evaluated points from the previous steps: \((-3, -9)\), \((0, 4)\), and \((3, 7)\).
* On graph paper or a digital graphing tool, plot these points accurately:
* \((-3, -9)\): This point is on the downward parabola corresponding to \(-x^2\) for \(x < 0\).
* \((0, 4)\): This point sits alone at \(x = 0\) on the graph.
* \((3, 7)\): This point falls on the line \(3x - 2\) for \(x > 0\).

Ensure lines are smooth and consistent with the behavior of the functions they represent. Each piece must be drawn distinctly if they switch behavior at boundary points like 0 in this case.

function evaluation

Function evaluation involves determining the output \(f(x)\) for a given input \(x\). It’s important to:

* Know which part of the piecewise function applies to your input \(x\).
* Substitute the input \(x\) directly into the appropriate formula.

Using our problem as an example, we evaluated:\(f(-3)\), \(f(0)\), and \(f(3)\) using the corresponding pieces:

* For \(x < 0\), apply \(-x^2\).
* For \(x = 0\), the function is simply \(4\).
* For \(x > 0\), apply \(3x - 2\).

Practicing function evaluation with different inputs will help in mastering piecewise functions.