(1)) Find all values of $$ k $$ that make $$ f(x) $$ continuous at $$ x=1 $$ $$ f(x)=\left\{\begin{array}{ll}-x^{2}-3 & \text { if } x \leq 1 \ -5 k x+k^{2} & \text { if } x1\end{array}\right. $$ $$ k= $$ Write your answer as a single value or a list of values separated by commas.

Question

(1)) Find all values of $$ k $$ that make $$ f(x) $$ continuous at $$ x=1 $$ $$ f(x)=\left\{\begin{array}{ll}-x^{2}-3 & \text { if } x \leq 1 \ -5 k x+k^{2} & \text { if } x>1\end{array}\right. $$ $$ k= $$ Write your answer as a single value or a list of values separated by commas.

UpStudy AI · Accepted Answer

To find the values of $$ k $$ that make the function $$ f(x) $$ continuous at $$ x=1 $$, we need to ensure that the left-hand limit and the right-hand limit of $$ f(x) $$ as $$ x $$ approaches 1 are equal.

Given:
$$ f(x)=\left\{\begin{array}{ll}-x^{2}-3 & \text { if } x \leq 1 \ -5 k x+k^{2} & \text { if } x>1\end{array}\right. $$

To find the left-hand limit:
$$ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (-x^{2}-3) = -1^{2}-3 = -4 $$

To find the right-hand limit:
$$ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (-5kx+k^{2}) = -5k(1)+k^{2} = -5k+k^{2} $$

For the function to be continuous at $$ x=1 $$, the left-hand limit and the right-hand limit must be equal. Therefore, we set the two limits equal to each other:
$$ -4 = -5k+k^{2} $$

Now, we can solve this equation to find the values of $$ k $$ that make the function continuous at $$ x=1 $$.
Solve the quadratic equation by following steps:
- step0: Solve by factoring:
  $$-4=-5k+k^{2}$$
- step1: Swap the sides:
  $$-5k+k^{2}=-4$$
- step2: Move the expression to the left side:
  $$-5k+k^{2}+4=0$$
- step3: Factor the expression:
  $$\left(k-4\right)\left(k-1\right)=0$$
- step4: Separate into possible cases:
  $$\begin{align}&k-4=0\&k-1=0\end{align}$$
- step5: Solve the equation:
  $$\begin{align}&k=4\&k=1\end{align}$$
- step6: Rewrite:
  $$k_{1}=1,k_{2}=4$$
The values of $$ k $$ that make the function $$ f(x) $$ continuous at $$ x=1 $$ are $$ k=1 $$ and $$ k=4 $$.
$$ k=1,4 $$

(1)) Find all values of that make continuous at

Write your answer as a single value or a list of values separated by commas.

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