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Math  /  Algebra

QuestionSPuS P u Q.13) If f(x)=sec⁡xx+3f(x)=\frac{\sec x}{x+3} and f−1(c)=0f^{-1}(c)=0, then c=c= A. 13\frac{1}{3} B. 0 C. 23\frac{2}{\sqrt{3}} D. 14\frac{1}{4} E.None

Studdy Solution

STEP 1

1. We are given the function f(x)=sec⁡xx+3 f(x) = \frac{\sec x}{x+3} .
2. We know that f−1(c)=0 f^{-1}(c) = 0 .
3. We need to find the value of c c .

STEP 2

1. Understand the meaning of f−1(c)=0 f^{-1}(c) = 0 .
2. Determine the value of x x for which f(x)=c f(x) = c .
3. Substitute x=0 x = 0 into the function f(x) f(x) .
4. Solve for c c .

STEP 3

Understand that f−1(c)=0 f^{-1}(c) = 0 means that when x=0 x = 0 , the function f(x) f(x) outputs c c .

STEP 4

Substitute x=0 x = 0 into the function f(x) f(x) :
f(0)=sec⁡(0)0+3 f(0) = \frac{\sec(0)}{0+3}

STEP 5

Calculate sec⁡(0) \sec(0) :
sec⁡(0)=1cos⁡(0)=1 \sec(0) = \frac{1}{\cos(0)} = 1

STEP 6

Substitute the value of sec⁡(0) \sec(0) into the equation:
f(0)=13 f(0) = \frac{1}{3}

STEP 7

Since f(0)=c f(0) = c , we have:
c=13 c = \frac{1}{3}
The value of c c is:
13 \boxed{\frac{1}{3}}

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