If the function
`f(x) = {(a+bx",", x lt 1), (5",", x=1), (b-ax",", x gt 1):}`
is continuous, then what is the value of `(a+b)`?

Options

A. `5`
B. `10`
C. `15`
D. `20`

Answer

To determine the value of \( a + b \) for the given piecewise function \( f(x) \), we need to ensure that the function is continuous at \( x = 1 \). The function is defined as follows:

\[ f(x) = \begin{cases} a + bx & \text{if } x < 1 \\ 5 & \text{if } x = 1 \\ b - ax & \text{if } x > 1 \end{cases} \]

### Step 1: Find the limit as \( x \) approaches 1 from the left
We first calculate the left-hand limit as \( x \) approaches 1:

\[ \lim_{x \to 1^-} f(x) = a + b(1) = a + b \]

### Step 2: Find the limit as \( x \) approaches 1 from the right
Next, we calculate the right-hand limit as \( x \) approaches 1:

\[ \lim_{x \to 1^+} f(x) = b - a(1) = b - a \]

### Step 3: Set the limits equal to each other for continuity
For the function to be continuous at \( x = 1 \), the left-hand limit must equal the right-hand limit, and both must equal \( f(1) \):

\[ a + b = 5 \quad \text{(1)} \]
\[ b - a = 5 \quad \text{(2)} \]

### Step 4: Solve the equations
Now we have two equations:

1. \( a + b = 5 \)
2. \( b - a = 5 \)

From equation (2), we can express \( b \) in terms of \( a \):

\[ b = a + 5 \]

Now substitute this expression for \( b \) into equation (1):

\[ a + (a + 5) = 5 \]
\[ 2a + 5 = 5 \]
\[ 2a = 0 \]
\[ a = 0 \]

### Step 5: Find the value of \( b \)
Substituting \( a = 0 \) back into the equation for \( b \):

\[ b = 0 + 5 = 5 \]

### Step 6: Calculate \( a + b \)
Now we can find \( a + b \):

\[ a + b = 0 + 5 = 5 \]

Thus, the value of \( a + b \) is \( \boxed{5} \).