what value of c makes the function continuous for all x?
f(x)= {x^2-c, x<5
4x+2c, x>=5}

To determine the value of \( c \) that makes the function continuous for all \( x \), we need to ensure that the left-hand limit and the right-hand limit of the function at \( x = 5 \) are equal, and that they both equal the value of the function at \( x = 5 \).

The given function is:

\[
f(x) = 
\begin{cases}
x^2 - c, & \text{if } x < 5 \\
4x + 2c, & \text{if } x \geq 5
\end{cases}
\]

To ensure continuity at \( x = 5 \), we need:

\[
\lim_{x \to 5^-} f(x) = \lim_{x \to 5^+} f(x) = f(5).
\]

### Step 1: Find the left-hand limit as \( x \to 5 \).

For \( x < 5 \), the function is \( f(x) = x^2 - c \). So, the left-hand limit is:

\[
\lim_{x \to 5^-} f(x) = 5^2 - c = 25 - c.
\]

### Step 2: Find the right-hand limit as \( x \to 5 \).

For \( x \geq 5 \), the function is \( f(x) = 4x + 2c \). So, the right-hand limit is:

\[
\lim_{x \to 5^+} f(x) = 4(5) + 2c = 20 + 2c.
\]

### Step 3: Set the limits equal to each other.

To make the function continuous at \( x = 5 \), set the left-hand and right-hand limits equal:

\[
25 - c = 20 + 2c.
\]

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

\[
25 - 20 = 2c + c,
\]

\[
5 = 3c,
\]

\[
c = \frac{5}{3}.
\]

### Final Answer:

The value of \( c \) that makes the function continuous for all \( x \) is:

\[
\boxed{\frac{5}{3}}
\]

Would you like further details or have any questions? Here are some related questions to explore:

1. What does it mean for a function to be continuous at a point?
2. How do you determine the continuity of piecewise functions?
3. What happens if \( c \) is not \( \frac{5}{3} \)? What does the graph look like?
4. How can you check continuity for functions involving more complex expressions?
5. How would you approach this problem if the point of interest was different?

**Tip:** When solving continuity problems, always equate the left and right limits at the point of interest to find unknown constants.

What value of c makes the function continuous for all x?

f(x) = {x^2 - c, x < 5
4x + 2c, x >= 5}

Discover how to find the value of c that ensures the continuity of a piecewise function. This problem involves limits and the definition of continuity, and it is suitable for students in Grades 11-12. Step-by-step solution provided.

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