Question

(a) Find the domain of each function. (b) Locate any intercepts. (c) Graph each function. (d) Based on the graph, find the range. $$f(x)=\left\\{\begin{array}{ll}x+3 & \text { if } x<-2 \\\\-2 x-3 & \text { if } x \geq-2\end{array}\right.$$

Short answer

Domain: \([-\infty, \infty]\). Intercepts: \(y = -3\), \(x = -3, -\frac{3}{2}\). Range: \([-\infty, \infty)\).

Step-by-step solution

- Find the Domain

The domain of a function is the set of all possible input values (x-values) for the function. Since the given function is defined for all real numbers (with two piecewise parts), the domain is all real numbers, or \([-\infty, \infty]\).

- Locate Intercepts

To find the intercepts, determine where the function crosses the x-axis (x-intercepts) and the y-axis (y-intercepts). For y-intercepts: set x = 0 and solve \(f(0) = -2(0) - 3 = -3 \). To find x-intercepts, set f(x) = 0 for both pieces: \(x+3 = 0 \Rightarrow x = -3 \) (valid since -3 \(< -2\)) and \(-2x-3 = 0 \Rightarrow x = -\frac{3}{2} \) (valid since -\frac{3}{2} \( \geq -2\)).

- Graph the Function

Graph the function by plotting the two pieces: \(-\infty < x < -2 \: y = x + 3\) and \(-2 \leq x \leq\infty\: y = -2x - 3\). Include the boundary point x = -2 where the function changes.

- Find the Range

From the graph, determine the set of all possible output values (y-values). For \(-\infty < x < -2 \), \(-\infty < y < 1\). For \(-2 \leq x \leq\infty\), the output values \([-3, \infty)\). Combining these intervals, the range is \([-\infty, \infty)\).

Key concepts

Domain of a Function

The domain of a function is the set of all possible input values (or x-values) for which the function is defined. For the given piecewise function, it consists of two parts: \( x + 3 \) when \( x < -2 \) and \( -2x - 3 \) when \( x \geq -2 \). Since there are no restrictions like square roots of negative numbers or divisions by zero, the function is defined for all real numbers. Thus, the domain is all real numbers, often written as \( \left( -\infty, \infty \right) \).
Simply put, you can plug in any x-value and get a valid output. Whether x is a very high positive number, a very low negative number, or zero, the function will always give a corresponding y-value.

Intercepts

Intercepts are the points where the graph of the function crosses the x-axis and y-axis.
To find the y-intercept, set \( x = 0 \) and solve for the function:

• For \(f(x) = -2x - 3 \), substituting \( x = 0 \) gives \( f(0) = -3 \). Hence, the y-intercept is \( (0, -3) \).

To find the x-intercept(s), set \( f(x) = 0 \) and solve for x for each part of the function:

• For \( x + 3 \):
  \( x + 3 = 0 \) ⟹ \( x = -3 \), and this is valid as -3 is less than -2.
  
• For \( -2x - 3 \):
  \( -2x - 3 = 0 \) ⟹ \( x = -\frac{3}{2} \), and this is valid as -1.5 is greater than or equal to -2.
  

So, the intercepts are \( (-3, 0) \) and \( \left( -\frac{3}{2}, 0 \right) \).

Graphing Functions

Graphing a piecewise function involves plotting each part of the function on the same set of axes. Here's how you do it step-by-step:

• For \( x < -2 \), plot the line \( y = x + 3 \). This is a straight line with a slope of 1 and a y-intercept of (0,3).
  
• For \( x \geq -2 \), plot the line \( y = -2x - 3 \). This is a straight line with a slope of -2 and a y-intercept of (0,-3).
  
• Ensure to plot the boundary point at \( x = -2 \) where the function changes. The value of the function at this point for both parts should match or be considered separately as a closed dot (if included) or an open dot (if not).
  

Connecting these lines gives us the graph of the piecewise function.

Range of a Function

The range of a function is the set of all possible output values (or y-values) that the function can produce. By examining the graph:

• For \( x < -2 \), the output values are from \( -\infty \) to 1, but not including 1, as the graph approaches 1 but doesn't reach it.
  
• For \( x \geq -2 \), the output values are from -7 to \( +\infty \), starting from -7 (not including) and going upwards.
  

Combining these, the continuous connection of the y-values from the piecewise sections means the range is \( \left( -\infty, \infty \right) \). This tells us that every possible y-value from negative infinity to positive infinity is included.