https://file.aiquickdraw.com/mathbot/file/17286372016193uwtwpbj.jpg

Find the value of 𝑘 so that 𝑔(𝑥) is continuous everywhere

To ensure that the function \( g(x) \) is continuous everywhere, the function must be continuous at the point where the two pieces of the piecewise function meet, which is at \( x = -2 \).

The given function is:
\[
g(x) =
\begin{cases} 
x^2 - 2x + 3, & \text{if } x \leq -2 \\
kx + 1, & \text{if } x > -2
\end{cases}
\]

For the function to be continuous at \( x = -2 \), the two pieces must meet smoothly at \( x = -2 \). This means the limit of the function from the left (\( x \to -2^- \)) must equal the limit from the right (\( x \to -2^+ \)) and both must equal the function value at \( x = -2 \).

### Step 1: Calculate \( g(-2) \) from the first part of the piecewise function
The left-hand side of the function is defined as \( g(x) = x^2 - 2x + 3 \) for \( x \leq -2 \). Substituting \( x = -2 \):
\[
g(-2) = (-2)^2 - 2(-2) + 3 = 4 + 4 + 3 = 11
\]
Thus, \( g(-2) = 11 \).

### Step 2: Find the right-hand limit as \( x \to -2^+ \)
For \( x > -2 \), the function is \( g(x) = kx + 1 \). Taking the limit as \( x \) approaches \(-2\) from the right:
\[
\lim_{x \to -2^+} g(x) = k(-2) + 1 = -2k + 1
\]

### Step 3: Set the two limits equal to ensure continuity
For continuity at \( x = -2 \), the function values must be equal from both sides:
\[
g(-2) = \lim_{x \to -2^+} g(x)
\]
\[
11 = -2k + 1
\]

### Step 4: Solve for \( k \)
\[
11 = -2k + 1
\]
\[
11 - 1 = -2k
\]
\[
10 = -2k
\]
\[
k = -5
\]

### Conclusion:
The value of \( k \) that makes the function \( g(x) \) continuous everywhere is \( \boxed{-5} \).

Would you like more details or have any further questions?

Here are 5 follow-up questions related to this problem:
1. How do we check if a function is differentiable at a point?
2. What happens if \( k \) had been a different value?
3. How would the graph of this piecewise function look like with \( k = -5 \)?
4. Can you explain the concept of continuity in a formal way?
5. What are the consequences if a function is not continuous at a point?

**Tip:** Continuity ensures that there are no jumps or breaks in a function at a particular point, which is crucial in calculus, especially for differentiation.

Discover how to find the value of k to make the piecewise function g(x) continuous everywhere. This detailed solution covers continuity concepts, limit calculations, and piecewise function rules, ideal for Grade 11-12 Pre-Calculus or Calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation