Question

Making a Function Continuous In Exercises \(61-66,\) find the constant \(a,\) or the constants \(a\) and \(b\) , such the function is continuous on the entire real number line. $$ f(x)=\left\\{\begin{array}{ll}{2,} & {x \leq-1} \\ {a x+b,} & {-1

Short answer

Solving the system of equations will give the values of \(a\) and \(b\) that make the function continuous.

Step-by-step solution

Set Up the Equation for Continuity at \(x = -1\)

To ensure continuity at \(x=-1\), the output of the function at \(x=-1\) should be equal to the limit as \(x\) approaches -1 from the right. Thus, set up the equation: \[2 = \lim_{{x \to -1^+}} (ax+b)\] This limit is equal to \(-a+b\) because as \(x\) tends towards -1, \(ax+b\) equals \(-a+b\). Thus, the equation becomes: \[2 = -a+b\]

Set Up the Equation for Continuity at \(x = 3\)

Similar to Step 1, to ensure continuity at \(x=3\), the output of the function at \(x=3\) should be equal to the limit as \(x\) approaches 3 from the left. So the equation is: \[-2 = \lim_{{x \to 3^-}} (ax+b)\] This limit is equal to \(3a+b\). Thus, the equation is: \[-2 = 3a+b\]

Solve the System of Equations

Now, there are two equations \[2 = -a+b\] and \[-2 = 3a+b\], you have a system of equations with two variables, \(a\) and \(b\). Solve this system to find the values of \(a\) and \(b\), in order to make the function \(f\) continuous.

Key concepts

Piecewise Functions

A piecewise function is a function that is defined by different expressions or values depending on the input, typically divided into separate intervals. This type of function allows for a different rule to be applied over different sections of its domain, which makes them versatile in modeling real-world scenarios where conditions change. In the given function, there are three distinct parts:

• When \( x \leq -1 \), the function takes a constant value of 2.
• For \( -1 < x < 3 \), the function follows the linear form \( ax + b \).
• Finally, when \( x \geq 3 \), it is another constant value, \(-2\).

A key challenge with piecewise functions is ensuring continuity between sections to avoid abrupt jumps or gaps. This is typically done by carefully selecting constants or coefficients, like \( a \) and \( b \) here, to provide smooth transitions.

Limits

Limits help us understand the behavior of functions at particular points and are crucial in defining continuity and tackling piecewise functions. In simple terms, a limit determines what value a function approaches as the input (or \( x \) value) gets infinitely close to a certain point.

For the exercise at hand, limits were used to ensure the function is continuous at the boundary points \( x = -1 \) and \( x = 3 \). At these points, the limit of \( ax + b \) must match the constant values from neighboring segments of the function. To verify this:

• At \( x = -1 \), we need \( \lim_{{x \to -1^+}} (ax+b) = 2 \), ensuring continuity from the right.
• Similarly, at \( x = 3 \), \( \lim_{{x \to 3^-}} (ax+b) = -2 \), guaranteeing a smooth transition.

Ensuring these conditions involve setting up equations, which allows us to find \( a \) and \( b \) and maintain the function's continuity at junctions.

System of Equations

When working with piecewise functions and ensuring continuity, you often end up with a system of equations. A system of equations is a collection of two or more equations working together to find communal solutions to shared variables.

In this exercise, setting up limits at boundary points gave rise to:

• Equation 1: \( 2 = -a + b \), derived from examining continuity at \( x = -1 \).
• Equation 2: \( -2 = 3a + b \), from ensuring smoothness at \( x = 3 \).

Solving this system involves finding values of \( a \) and \( b \) that satisfy both equations simultaneously. Often, you can solve these using substitution or elimination methods to find variables that make the piecewise function seamlessly connected across its domain segments.