Math Problem Statement

Tentukan invers dari fungsi berikut dan selidiki mana yang merupakan fungsi invers: a) f = {(a,b), (b,c), (c,d), (d,e)}, b) g = {(x,-2), (0,2), (1,-1), (2,-3)}, c) h = {(-1,6), (0,-1), (2,2), (6,0)}

Solution

The question asks to find the inverse of the given functions and determine which of them represent actual inverse functions. There are three relations defined by their ordered pairs:

  1. f={(a,b),(b,c),(c,d),(d,e)}f = \{(a,b), (b,c), (c,d), (d,e)\}
  2. g={(x,2),(0,2),(1,1),(2,3)}g = \{(x,-2), (0,2), (1,-1), (2,-3)\}
  3. h={(1,6),(0,1),(2,2),(6,0)}h = \{(-1,6), (0,-1), (2,2), (6,0)\}

Steps to find the inverse:

The inverse of a function f(x)f(x) is found by swapping the input-output pairs, i.e., if f={(x1,y1),(x2,y2),}f = \{(x_1, y_1), (x_2, y_2), \dots \}, then the inverse f1f^{-1} is {(y1,x1),(y2,x2),}\{(y_1, x_1), (y_2, x_2), \dots \}.

Inverses of each function:

a) f1f^{-1} would be: f1={(b,a),(c,b),(d,c),(e,d)}f^{-1} = \{(b,a), (c,b), (d,c), (e,d)\} Here, the function seems well-behaved, but for it to be a proper function, all the yy-values (which become the new xx-values) must be unique.

b) g1g^{-1} would be: g1={(2,x),(2,0),(1,1),(3,2)}g^{-1} = \{(-2,x), (2,0), (-1,1), (-3,2)\} This also appears to be a valid function with distinct yy-values (now xx-values).

c) h1h^{-1} would be: h1={(6,1),(1,0),(2,2),(0,6)}h^{-1} = \{(6,-1), (-1,0), (2,2), (0,6)\} Again, this forms a function because the new xx-values are unique.

Determining which is a valid inverse function:

  • A function is invertible if no two different inputs produce the same output (i.e., it passes the horizontal line test for invertibility). Checking this:
    • Function ff: May fail the invertibility test depending on the relation between a,b,c,d,ea, b, c, d, e, but as shown here, we can't confirm.
    • Function gg: This is likely invertible since each input maps to a unique output.
    • Function hh: Also seems invertible.

Conclusion:

Functions gg and hh are likely to be valid inverse functions, but for ff, additional clarification about the values would be needed.


Would you like more details or have any questions? Here are five related questions to expand the topic:

  1. How do we verify if a function is one-to-one and invertible?
  2. What properties must a function satisfy to have an inverse?
  3. Can you explain the horizontal line test and its relation to invertibility?
  4. How do we graphically represent the inverse of a function?
  5. Are there cases where functions have partial inverses?

Tip: Always check if a function is one-to-one before finding its inverse, as only one-to-one functions have true inverses.

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Relations

Formulas

If f = {(x1, y1), (x2, y2), ...}, then f^(-1) = {(y1, x1), (y2, x2), ...}

Theorems

Inverse function properties: One-to-one correspondence, uniqueness of output.

Suitable Grade Level

Grades 10-12