Find all values of c such that f is continuous on (−∞,∞).f(x)={1−x2,x≤c {x,x&gt;c

Question

Find all values of c such that f is continuous on (−∞,∞).f(x)={1−x2,x≤c {x,x&amp;gt;c

Robert Harris · Accepted Answer

Since&amp;nbsp;1−x2&amp;nbsp;and x are continuous on&amp;nbsp;(−∞,∞), the only possible point of discontinuity is at x=c. Right now, both functions ought to be equal in value, so1−x2=x&amp;nbsp;x2+x−1=0&amp;nbsp;Therefore, using quadratic formula,&amp;nbsp;x=±5−12&amp;nbsp;So the given function is continuous for&amp;nbsp;c=±5−12&amp;nbsp;Result:&amp;nbsp;c=±5−12

Eliza Beth13 · Accepted Answer

To find all values of c such that f is continuous on (-∞, ∞) for the function

f (x) = {\begin{matrix} 1 - x^{2} & x \leq c \\ x & x > c \end{matrix}

, we need to consider the conditions for continuity at the point c.
For f to be continuous at c, the left-hand limit and right-hand limit must exist and be equal to the function value at c.
Let's calculate these limits and analyze the cases:
Case 1:

x \leq c

In this case, the function is given by f(x) = 1-x^2.

{lim}_{x \to c^{-}} f (x) = {lim}_{x \to c^{-}} (1 - x^{2}) = 1 - c^{2}

Case 2: x > c
In this case, the function is given by f(x) = x.

{lim}_{x \to c^{+}} f (x) = {lim}_{x \to c^{+}} x = c

For f to be continuous, the left-hand limit (Case 1) must equal the right-hand limit (Case 2) at c.
Therefore, we have the equation:

1 - c^{2} = c

Now, let's solve this equation to find the values of c:

1 - c^{2} = c

c^{2} + c - 1 = 0

Using the quadratic formula:

c = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

a = 1

,

b = 1

, and

c = - 1

c = \frac{- 1 \pm \sqrt{1^{2} - 4 \cdot 1 \cdot (- 1)}}{2 \cdot 1}

c = \frac{- 1 \pm \sqrt{1 + 4}}{2}

c = \frac{- 1 \pm \sqrt{5}}{2}

Hence, the values of c such that f is continuous on (-∞, ∞) are:

c = \frac{- 1 + \sqrt{5}}{2}

and

c = \frac{- 1 - \sqrt{5}}{2}

Don Sumner · Accepted Answer

Answer:−1+52,−1−52Explanation:To find all values of c such that f is continuous on (−∞,∞), we need to check the continuity of f at c.First, let&#039;s consider the left limit of f at c:limx→c−f(x)=limx→c−(1−x2)=1−c2Next, let&#039;s consider the right limit of f at c:limx→c+f(x)=limx→c+x=cFor f to be continuous at c, we need these two limits to be equal. Therefore, we must have:1−c2=cSolving this equation, we get:c2+c−1=0Using the quadratic formula, we find:c=−1±52Therefore, the values of c that make f continuous on (−∞,∞) are:c=−1+52,−1−52

madeleinejames20 · Accepted Answer

Step 1:To find all values of c such that f is continuous on (−∞,∞), we need to ensure that the two pieces of f(x) match at the point c. This means that the left-hand limit and the right-hand limit of f(x) as x approaches c must be equal.Let&#039;s calculate the left-hand limit and the right-hand limit separately:Left-hand limit as x approaches c:limx→c−f(x)=limx→c−(1−x2)=1−c2Right-hand limit as x approaches c:limx→c+f(x)=limx→c+x=cStep 2:For f to be continuous at c, the left-hand limit and the right-hand limit must be equal. Therefore, we have the equation:1−c2=cTo solve this equation, we can rearrange it:c2+c−1=0We can now solve this quadratic equation. Using the quadratic formula, we get:c=−1±12−4(1)(−1)2(1)Simplifying further:c=−1±1+42c=−1±52Hence, the values of c for which f is continuous on (−∞,∞) are:c=−1+52andc=−1−52

Find all values of c such that f is continuous

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