Math Problem Statement

Tentukan fungsi invers berikut: 1. f(x) = 3x - 5, 2. f(x) = 5x + 7, 3. f(x) = (x + 5)/(2x - 6), 4. f(x) = (2x + 1)/(8x - 5), 5. f(x) = (x + 1)/(7 - 6x).

Solution

Let's go through each function step by step to find their inverses.

  1. f(x)=3x5f(x) = 3x - 5:

    • To find the inverse, first replace f(x)f(x) with yy:
      y=3x5y = 3x - 5
    • Switch xx and yy:
      x=3y5x = 3y - 5
    • Solve for yy:
      y=x+53y = \frac{x + 5}{3}
    • So, the inverse is f1(x)=x+53f^{-1}(x) = \frac{x + 5}{3}.
  2. f(x)=5x+7f(x) = 5x + 7:

    • Replace f(x)f(x) with yy:
      y=5x+7y = 5x + 7
    • Switch xx and yy:
      x=5y+7x = 5y + 7
    • Solve for yy:
      y=x75y = \frac{x - 7}{5}
    • So, the inverse is f1(x)=x75f^{-1}(x) = \frac{x - 7}{5}.
  3. f(x)=x+52x6f(x) = \frac{x + 5}{2x - 6}:

    • Replace f(x)f(x) with yy:
      y=x+52x6y = \frac{x + 5}{2x - 6}
    • Switch xx and yy:
      x=y+52y6x = \frac{y + 5}{2y - 6}
    • Solve for yy by cross-multiplying:
      x(2y6)=y+5x(2y - 6) = y + 5
      2xy6x=y+52xy - 6x = y + 5
    • Group the terms with yy:
      2xyy=6x+52xy - y = 6x + 5
      y(2x1)=6x+5y(2x - 1) = 6x + 5
    • Solve for yy:
      y=6x+52x1y = \frac{6x + 5}{2x - 1}
    • So, the inverse is f1(x)=6x+52x1f^{-1}(x) = \frac{6x + 5}{2x - 1}.
  4. f(x)=2x+18x5f(x) = \frac{2x + 1}{8x - 5}:

    • Replace f(x)f(x) with yy:
      y=2x+18x5y = \frac{2x + 1}{8x - 5}
    • Switch xx and yy:
      x=2y+18y5x = \frac{2y + 1}{8y - 5}
    • Solve for yy by cross-multiplying:
      x(8y5)=2y+1x(8y - 5) = 2y + 1
      8xy5x=2y+18xy - 5x = 2y + 1
    • Group the terms with yy:
      8xy2y=5x+18xy - 2y = 5x + 1
      y(8x2)=5x+1y(8x - 2) = 5x + 1
    • Solve for yy:
      y=5x+18x2y = \frac{5x + 1}{8x - 2}
    • So, the inverse is f1(x)=5x+18x2f^{-1}(x) = \frac{5x + 1}{8x - 2}.
  5. f(x)=x+176xf(x) = \frac{x + 1}{7 - 6x}:

    • Replace f(x)f(x) with yy:
      y=x+176xy = \frac{x + 1}{7 - 6x}
    • Switch xx and yy:
      x=y+176yx = \frac{y + 1}{7 - 6y}
    • Solve for yy by cross-multiplying:
      x(76y)=y+1x(7 - 6y) = y + 1
      7x6xy=y+17x - 6xy = y + 1
    • Group the terms with yy:
      7x1=y+6xy7x - 1 = y + 6xy
      7x1=y(1+6x)7x - 1 = y(1 + 6x)
    • Solve for yy:
      y=7x11+6xy = \frac{7x - 1}{1 + 6x}
    • So, the inverse is f1(x)=7x11+6xf^{-1}(x) = \frac{7x - 1}{1 + 6x}.

Would you like more details on any step? Here are some related questions:

  1. What is the general process for finding the inverse of a function?
  2. How can we verify that two functions are inverses of each other?
  3. Why is switching xx and yy a key step in finding the inverse?
  4. What kind of functions do not have inverses, and why?
  5. How do we deal with more complex functions, such as those involving higher-degree polynomials, when finding inverses?

Tip: Always check if a function is one-to-one before finding its inverse, as non-invertible functions cannot have proper inverses.

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Algebra

Formulas

To find the inverse, switch x and y, then solve for y.

Theorems

Inverse Function Theorem: For a function to have an inverse, it must be one-to-one and onto (bijective).

Suitable Grade Level

Grades 10-12