Understanding Continuity of Piecewise Functions

A function \( f(x) \) is said to be continuous at a point \( c \) if the following three conditions are met:

1. The function \( f(c) \) is defined.
2. The limit of the function as \( x \) approaches \( c \) exists (i.e., \( \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) \)).
3. The limit of the function equals the function value at the point: \( \lim_{x \to c} f(x) = f(c) \).

For a piecewise function like the one given, we need to ensure continuity at the points where the definition of the function changes. In this problem, the function definition changes at \( x = 1 \) and \( x = 5 \). To make the function \( f(x) \) continuous everywhere, it must be continuous at these critical points.

Checking Continuity at \( x = 1 \)

For \( f(x) \) to be continuous at \( x = 1 \), the left-hand limit, the right-hand limit, and the function value at \( x = 1 \) must all be equal.

• Left-hand limit at \( x = 1 \): We use the definition \( f(x) = 3 \) for \( x \leq 1 \).
  \( \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} 3 = 3 \)
• Function value at \( x = 1 \): We use the definition \( f(x) = 3 \) for \( x \leq 1 \).
  \( f(1) = 3 \)
• Right-hand limit at \( x = 1 \): We use the definition \( f(x) = ax + b \) for \( 1 < x < 5 \).
  \( \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (ax + b) = a(1) + b = a + b \)

For continuity at \( x = 1 \), we must have: \( \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1) \).
This gives us the equation: \( 3 = a + b \). Let's call this Equation (1):

\( a + b = 3 \quad \cdots (1) \)

Checking Continuity at \( x = 5 \)

For \( f(x) \) to be continuous at \( x = 5 \), the left-hand limit, the right-hand limit, and the function value at \( x = 5 \) must all be equal.

• Left-hand limit at \( x = 5 \): We use the definition \( f(x) = ax + b \) for \( 1 < x < 5 \).
  \( \lim_{x \to 5^-} f(x) = \lim_{x \to 5^-} (ax + b) = a(5) + b = 5a + b \)
• Function value at \( x = 5 \): We use the definition \( f(x) = 10 \) for \( x \geq 5 \).
  \( f(5) = 10 \)
• Right-hand limit at \( x = 5 \): We use the definition \( f(x) = 10 \) for \( x \geq 5 \).
  \( \lim_{x \to 5^+} f(x) = \lim_{x \to 5^+} 10 = 10 \)

For continuity at \( x = 5 \), we must have: \( \lim_{x \to 5^-} f(x) = \lim_{x \to 5^+} f(x) = f(5) \).
This gives us the equation: \( 5a + b = 10 \). Let's call this Equation (2):

\( 5a + b = 10 \quad \cdots (2) \)

Solving for a and b

We now have a system of two linear equations with two variables, \( a \) and \( b \):

1. \( a + b = 3 \)
2. \( 5a + b = 10 \)

We can solve this system using various methods, such as substitution or elimination. Let's use elimination by subtracting Equation (1) from Equation (2):

\( (5a + b) - (a + b) = 10 - 3 \)

\( 5a + b - a - b = 7 \)

\( 4a = 7 \)

\( a = \frac{7}{4} \)

Now substitute the value of \( a \) into Equation (1) to find \( b \):

\( \frac{7}{4} + b = 3 \)

\( b = 3 - \frac{7}{4} \)

\( b = \frac{12}{4} - \frac{7}{4} \)

\( b = \frac{5}{4} \)

Finding the Sum of a and b

The question asks for the sum of the values of \( a \) and \( b \). We found \( a = \frac{7}{4} \) and \( b = \frac{5}{4} \).

\( a + b = \frac{7}{4} + \frac{5}{4} \)

\( a + b = \frac{7 + 5}{4} \)

\( a + b = \frac{12}{4} \)

\( a + b = 3 \)

Thus, the sum of the values of \( a \) and \( b \) that make the function continuous is 3.

Revision Table: Steps to Ensure Piecewise Function Continuity

Additional Information: Properties of Continuous Functions

If functions \( f \) and \( g \) are continuous at a point \( c \), then the following functions are also continuous at \( c \):

• Sum: \( (f+g)(x) = f(x) + g(x) \)
• Difference: \( (f-g)(x) = f(x) - g(x) \)
• Product: \( (fg)(x) = f(x)g(x) \)
• Constant Multiple: \( (kf)(x) = kf(x) \) for any constant \( k \)
• Quotient: \( (\frac{f}{g})(x) = \frac{f(x)}{g(x)} \), provided \( g(c) \neq 0 \)
• Composition: If \( g \) is continuous at \( c \) and \( f \) is continuous at \( g(c) \), then the composite function \( (f \circ g)(x) = f(g(x)) \) is continuous at \( c \).

Polynomial functions, exponential functions, sine, cosine, and absolute value functions are continuous everywhere on their domains.