Problem 46 - College Algebra and Calculus: An Applied Approach

Question

Find the constant \(a\) (Exercise 45\()\) and the constants \(a\) and \(b\) (Exercise 46 ) such that the function is continuous on the entire real line. \(f(x)=\left\\{\begin{array}{ll}2, & x \leq-1 \\ a x+b, & -1

Step-by-Step Solution

Answer

Here, \(a = -1\) and \(b = 1\).

Step 1: Determine the function behavior at x = -1

Set the value of x to -1 in the last two function formulae and equate them, because the function has to be equal in two adjacent intervals at the breaking points. This gives \(a(-1) + b = 2\). This is the first equation to solve for the variables a and b.

Step 2: Determine the function behavior at x = 3

Set the value of x to 3 in the last two function formulae and equate them. This gives \(a(3) + b = -2\). This is the second equation to solve for the variables a and b.

Step 3: Find the solution for 'a' and 'b'

To solve for a and b, we now have a system of two equations. Subtract the second equation from the first one, we get \(a(-1 - 3) = 2 - (-2)\), from which we get \(a = \frac{4}{-4} = -1\). Substitute a = -1 into the first equation, we get \(-1(-1) + b = 2\), then \(b = 2 - 1 = 1\).

Key Concepts

Piecewise Functions

Piecewise functions are a special type of function composed of multiple sub-functions, each of which applies to a particular interval of the main function's domain. These sub-functions are designed such that their outputs align and match at specific points, commonly known as boundary or transition points. Understanding piecewise functions is pivotal to grasping how a function behaves across different intervals, looking very much like a segmented structure working under a set of defined rules for each segment.

A key concept to grasp is continuity at the breakpoints where sub-functions meet. For a piecewise function to be continuous, the output or value from one part of a function must match the output from the adjacent function at the given transition points.

For example, if you're given a piecewise function like this:

• For \(x \leq -1\), \(f(x) = 2\)
• For \(-1 < x < 3\), \(f(x) = ax + b\)
• For \(x \geq 3\), \(f(x) = -2\)

Matching the values at the boundary points ensures the function does not "jump" where the segments of the piecewise function meet. This is important so that the function can be called continuous.

System of Equations

A system of equations is a collection of two or more equations with a same set of unknowns. In mathematics, these systems are used to find values of unknown variables that satisfy all the equations in the system simultaneously.

In the context of making piecewise functions continuous, we often derive equations based on the condition that function values meet neatly at the transition points. For instance, for the piecewise function we are analyzing, we derived:

• At \(x = -1\), we form equation 1: \(a(-1) + b = 2\)
• At \(x = 3\), equation 2 arises: \(a(3) + b = -2\)

These two form a linear system of equations. By solving this system simultaneously, we derive the values for constants \(a\) and \(b\). The method involves algebraic manipulations such as adding or subtracting equations from each other to eliminate variables, ultimately finding the solution.

Function Behavior at Points

Understanding the behavior of functions at specific points, particularly boundaries and transition points, is incredibly crucial when dealing with piecewise functions. The behavior of a function at a particular point tells us how the function's output is transitioning from one sub-function to another as the input crosses boundaries.

In our example, continuity means the function's behavior at \(x = -1\) and \(x = 3\) needs special attention. At these points, \(f(x)\) must provide the same output value not only from the left (i.e., limits approaching the point from one side) but also from the right of each boundary point. Thus:

• Ensure the value from the sub-function on one side matches the value on the other sub-function at \(x = -1\) and \(x = 3\).
• Check that left-hand limits and right-hand limits both equal the function value at these points to ensure continuity.

Adjusting individual parts of the function, often using constants like \(a\) and \(b\), enables a smooth, seamless transition from one piece of the function to the next at those crucial points.