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Question: Consider a function f: $(-\infty, 20] \rightarrow [0, \infty)$ defined as $f(x) = |x - 20|$. Then th...

Consider a function f: (−∞,20]→[0,∞)(-\infty, 20] \rightarrow [0, \infty) defined as f(x)=∣x−20∣f(x) = |x - 20|. Then the number of integral solutions of the equation f(x)=f−1(x)f(x) = f^{-1}(x) is

Answer

21

Explanation

Solution

To find the number of integral solutions of the equation f(x)=f−1(x)f(x) = f^{-1}(x), we first need to determine the explicit form of f(x)f(x) for its given domain, then find its inverse f−1(x)f^{-1}(x) along with its domain.

  1. Determine the function f(x)f(x):

    The function is given as f(x)=∣x−20∣f(x) = |x - 20| with the domain (−∞,20](-\infty, 20]. For any x∈(−∞,20]x \in (-\infty, 20], we have x−20≤0x - 20 \le 0. Therefore, ∣x−20∣=−(x−20)=20−x|x - 20| = -(x - 20) = 20 - x. So, f(x)=20−xf(x) = 20 - x for x∈(−∞,20]x \in (-\infty, 20].

  2. Determine the range of f(x)f(x):

    As xx approaches −∞-\infty, f(x)=20−xf(x) = 20 - x approaches ∞\infty. When x=20x = 20, f(20)=20−20=0f(20) = 20 - 20 = 0. Since f(x)=20−xf(x) = 20 - x is a decreasing function, its range is [f(20),lim⁡x→−∞f(x))=[0,∞)[f(20), \lim_{x \to -\infty} f(x)) = [0, \infty). So, the range of ff is [0,∞)[0, \infty).

  3. Determine the inverse function f−1(x)f^{-1}(x):

    Let y=f(x)=20−xy = f(x) = 20 - x. To find the inverse, we swap xx and yy: x=20−yx = 20 - y. Now, solve for yy: y=20−xy = 20 - x. So, f−1(x)=20−xf^{-1}(x) = 20 - x.

  4. Determine the domain and range of f−1(x)f^{-1}(x):

    The domain of f−1f^{-1} is the range of ff, which is [0,∞)[0, \infty). The range of f−1f^{-1} is the domain of ff, which is (−∞,20](-\infty, 20]. Thus, f−1(x)=20−xf^{-1}(x) = 20 - x for x∈[0,∞)x \in [0, \infty).

  5. Solve the equation f(x)=f−1(x)f(x) = f^{-1}(x):

    We need to solve 20−x=20−x20 - x = 20 - x. This equation is an identity, meaning it is true for all values of xx for which both f(x)f(x) and f−1(x)f^{-1}(x) are defined. The domain of f(x)f(x) is Df=(−∞,20]D_f = (-\infty, 20]. The domain of f−1(x)f^{-1}(x) is Df−1=[0,∞)D_{f^{-1}} = [0, \infty). For the equation f(x)=f−1(x)f(x) = f^{-1}(x) to be valid, xx must be in the intersection of their domains: x∈Df∩Df−1=(−∞,20]∩[0,∞)=[0,20]x \in D_f \cap D_{f^{-1}} = (-\infty, 20] \cap [0, \infty) = [0, 20].

  6. Find the number of integral solutions:

    The solutions to the equation f(x)=f−1(x)f(x) = f^{-1}(x) are all xx in the interval [0,20][0, 20]. We are looking for integral solutions, which are the integers in this interval. These integers are 0,1,2,…,200, 1, 2, \dots, 20. The number of integral solutions is 20−0+1=2120 - 0 + 1 = 21.