https://file.aiquickdraw.com/mathbot/file/17262143454858ifyzes0.jpg

The problem asks to determine the constants \(a\) and \(b\) so that the function \(f(x)\) is continuous over the interval \([0, \infty)\).

The function is defined piecewise as:
\[
f(x) = \begin{cases} 
\frac{ax^2 + b}{x^2 - 4}, & 0 \leq x < 2 \\
x^2 - 3x + 5, & x \geq 2
\end{cases}
\]

To ensure the function is continuous at \(x = 2\), the two pieces of the function must have the same value at \(x = 2\). This gives us a condition for continuity.

### Step 1: Find \(f(2)\) for both expressions
1. **For \(x \geq 2\):**
   \[
   f(2) = 2^2 - 3(2) + 5 = 4 - 6 + 5 = 3
   \]

2. **For \(0 \leq x < 2\):**
   \[
   f(2) = \frac{a(2)^2 + b}{(2)^2 - 4} = \frac{4a + b}{4 - 4} = \frac{4a + b}{0}
   \]
   Since dividing by zero is undefined, the numerator must also be zero for the expression to be valid. This gives us the condition:
   \[
   4a + b = 0
   \]
   (Equation 1)

### Step 2: Apply continuity condition
We know from the other piece of the function that at \(x = 2\), \(f(2) = 3\). Thus:
\[
f(2) = 3 \quad \text{for both pieces}.
\]

Therefore, the limit as \(x\) approaches 2 from the left must also be 3. Using the first part of the piecewise function:
\[
f(2) = 3 = \frac{4a + b}{0} \quad \text{(from left, at \(x = 2\))}
\]
We already know that \(4a + b = 0\), so from Equation 1:
\[
4a + b = 0
\]

### Step 3: Solve for \(a\) and \(b\)



Tentukanlah konstanta a dan b agar fungsi f(x) kontinu di selang [0, ∞): f(x) = {ax^2 + b / x^2 - 4, 0 ≤ x < 2; x^2 - 3x + 5, x ≥ 2}.

Learn how to determine the constants a and b to ensure the continuity of a piecewise function defined over the interval [0, ∞). The solution involves applying the continuity condition at x = 2 and solving the resulting equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation