Question

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. (Hint: Check for discontinuities.) Sketch the graph of the function. $$ y=\left\\{\begin{array}{ll} 3 x+1, & x \leq 1 \\ 5-x^{2}, & x>1 \end{array}\right. $$

Short answer

The function \( y = 3x + 1 \) is increasing on the interval \( x \leq 1 \), while the function \( y = 5 - x^2 \) is decreasing on the interval \( x > 1 \). There are no critical numbers. The graph of the function is a line with positive slope 3 for \( x \leq 1 \), and a downward-opening parabola for \( x > 1 \), with functions intersecting at the point \( (1,4) \).

Step-by-step solution

Differentiate the function

Given that the function is a piecewise function, differentiating each part gives: \[ y' = 3, x \leq 1 \], and \[ y' = -2x, x > 1 \]

Find the critical numbers

Critical number are points where the derivative is zero or undefined. By setting \( y' = 0 \) for each piece, we obtain that there's no critical point for \( 3x + 1 \) as its derivative is constant. For \( 5 - x^2 \), \( -2x = 0 \implies x = 0 \), however, this doesn't lie in the domain \( x > 1 \), hence no critical numbers.

Determine where the function is increasing or decreasing

Check where \( y' \) is positive or negative. For the piece \( 3x+1 \), \( y' = 3 > 0 \), thus this part of function is increasing on its domain \( x \leq 1 \). For the piece \( 5 - x^2 \), \( y' = -2x < 0 \), thus this function is decreasing on its domain \( x > 1 \).

Sketch the graph of the function

Graphing these functions using a graphic calculator or plotting software gives a line with positive slope 3 for \( x \leq 1 \) and a downward-opening parabola for \( x > 1 \). The functions intersect at the point \( (1,4) \).

Key concepts

Piecewise Function Differentiation

Differentiating piecewise functions like the provided exercise involves taking the derivative of each piece individually. These functions are defined by different expressions over different intervals. It's crucial to remember that the derivative is a measure of how a function changes as its input changes. This means that for piecewise functions, this rate of change might differ from one interval to another.

For the given function, for the interval where \( x \leq 1 \), the function is \( 3x + 1 \) and its derivative is a constant 3. This suggests a steady rate of change; in other words, for every unit increase in \( x \), \( y \) increases by 3 units. For the interval where \( x > 1 \), the function is \( 5 - x^2 \), and its derivative is \( -2x \), indicating that the rate of change in this interval depends on the value of \( x \).

The critical numbers of a function are found by setting the derivative equal to zero or finding where the derivative is undefined. These numbers are vital as they are potential locations of maximums, minimums, or points of inflection. It is important to highlight that critical numbers must lie within the domain of their respective pieces of the piecewise function.

Increasing and Decreasing Intervals

Understanding increasing and decreasing intervals of a function is essential for analyzing its behavior. An interval is increasing when the slope of the tangent lines to the curve is positive, meaning that as \( x \) increases, \( y \) also increases. Conversely, an interval is decreasing when the slope of the tangent lines to the curve is negative, indicating that as \( x \) increases, \( y \) decreases.

In the step by step solution, the derivative of the first piece, \( y' = 3 \), is always positive, thus the function is continuously increasing on the interval \( x \leq 1 \). On the other hand, for the second piece, \( y' = -2x \), the derivative is negative for all \( x > 1 \); therefore, the function is decreasing on that interval. Since the slope is not only negative but also varies with \( x \), the rate of decrease changes as well — it becomes more pronounced as \( x \) gets larger.

To better understand how these intervals affect the function, take the derivative's sign as an indicator. A positive derivative means upward movement, and a negative derivative means downward movement as you traverse the graph from left to right.

Graphing Piecewise Functions

Graphing piecewise functions may seem tricky at first, but understanding the unique traits of each piece allows you to piece together the overall shape of the graph. To sketch the graph effectively, it's helpful to note the behavior of the function on its increasing and decreasing intervals, and the points where the function's behavior changes — including discontinuities and junctions where the pieces connect.

In our example, graphing the first function \( 3x + 1 \) on the interval \( x \leq 1 \) gives a straight line with a slope of 3, intercepting the y-axis at 1. For \( x > 1 \), the second function produces a parabola that opens downward, starting from \( y = 4 \) when \( x = 1 \). The intersection point, \( (1, 4) \), is crucial because it's where the two pieces 'meet' on the graph. This is not a critical number but a junction point on the graph, and its identification is vital for a continuous, accurate graph representation.

Note the visual difference between increasing and decreasing intervals: the line segment with a constant positive slope up to the point \( x = 1 \) and the parabola that decreases thereafter. The union of these individual graphs yields the complete graph of the piecewise function. Such visual distinctions can often simplify understanding the function's behavior over its entire domain.