Find the value of the constant k that makes the function continuous.
​g(x)equalsStartSet Start 2 By 2 Matrix 1st Row 1st Column StartFraction 2 x squared minus 3 x minus 20 Over x minus 4 EndFraction 2nd Column if x not equals 4 2nd Row 1st Column kx minus 15 2nd Column if x equals 4 EndMatrix 
 


To make the function \( g(x) \) continuous, the left-hand limit as \( x \) approaches 4 from values other than 4 must equal the value of the function at \( x = 4 \).

### Given function:
\[
g(x) =
\begin{cases}
\frac{2x^2 - 3x - 20}{x - 4}, & \text{if } x \neq 4 \\
kx - 15, & \text{if } x = 4
\end{cases}
\]

### Step 1: Calculate the limit of the first piece as \( x \to 4 \).

The expression we are working with when \( x \neq 4 \) is:
\[
\frac{2x^2 - 3x - 20}{x - 4}
\]
First, factor the numerator \( 2x^2 - 3x - 20 \).

We want to factor \( 2x^2 - 3x - 20 \). Find two numbers that multiply to \( 2 \times (-20) = -40 \) and add to \( -3 \). These numbers are \( -8 \) and \( 5 \), so we can factor the quadratic as follows:

\[
2x^2 - 3x - 20 = (2x + 5)(x - 4)
\]

Thus, the expression becomes:

\[
\frac{(2x + 5)(x - 4)}{x - 4}
\]

For \( x \neq 4 \), we can cancel the \( x - 4 \) terms:

\[
\frac{(2x + 5)(x - 4)}{x - 4} = 2x + 5
\]

### Step 2: Find the limit as \( x \to 4 \).

Now, find the limit of \( 2x + 5 \) as \( x \to 4 \):

\[
\lim_{x \to 4} (2x + 5) = 2(4) + 5 = 8 + 5 = 13
\]

### Step 3: Set up the condition for continuity.

For the function to be continuous at \( x = 4 \), the value of \( g(x) \) at \( x = 4 \) must be equal to the limit as \( x \to 4 \) from the first piece. So, we require:

\[
kx - 15 = 13 \quad \text{when} \quad x = 4
\]

Substitute \( x = 4 \) into \( kx - 15 \):

\[
4k - 15 = 13
\]

### Step 4: Solve for \( k \).

\[
4k = 13 + 15
\]
\[
4k = 28
\]
\[
k = \frac{28}{4} = 7
\]

### Final Answer:
The value of the constant \( k \) that makes the function continuous is \( \boxed{7} \).

Would you like further details or have any questions?

### 5 related questions:
1. How do you check if a function is continuous at a point?
2. Can a function be continuous but not differentiable at a point?
3. What is the difference between continuity and differentiability?
4. How do you factor quadratics when the leading coefficient is not 1?
5. What are the conditions for applying L'Hopital's Rule in limit problems?

### Tip:
When checking for continuity at a point, ensure that the function’s limit from both sides equals the value of the function at that point.

Find the value of the constant k that makes the function continuous.
g(x) =
⎧
(2x^2 - 3x - 20) / (x - 4) if x ≠ 4
kx - 15 if x = 4
⎫

This detailed solution shows how to find the constant k that makes the piecewise function g(x) continuous at x = 4. It involves using the limit definition of continuity, factoring quadratics, and solving for k. Suitable for Grades 11-12.

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