Determine The Inverse Of The Function 3

determine the inverse of the function: 3.

The assignment requires us to find the inverse of a given function. The inverse of a function essentially “reverses” the original function, swapping the input and output values. To determine the inverse algebraically, we typically start with the original function expressed as y = f(x), then solve for x in terms of y, and finally interchange x and y to express the inverse function.

Let's assume the specific function to invert is given in the form f(x) = (ax + b)/(cx + d), which is a common linear fractional transformation. The goal is to find f-1(x). To do this, we follow these steps:

  1. Write y = f(x): y = (ax + b)/(cx + d).
  2. Swap variables: x = (ay + b)/(cy + d).
  3. Solve this equation for y:

Multiply both sides by (cy + d): x(cy + d) = ay + b.

Distribute x: xcy + xd = ay + b.

Group terms with y: xcy - ay = b - xd.

Factor y out: y(xc - a) = b - xd.

Finally, solve for y: y = (b - xd)/(xc - a).

Replacing y with f-1(x) yields:

f-1(x) = (b - xd)/(xc - a).

This general process demonstrates how to find the inverse algebraically for rational functions. The graphical approach involves reflecting the original graph over the line y = x, which can be visualized but is less exact without a graphing tool.

References

  • Larson, R., & Hostetler, R. (2013). Calculus with Applications. Cengage Learning.
  • Purcell, D., & Vette, R. (2011). Algebra and Trigonometry. Pearson.
  • Blitzer, R. (2014). Algebra and Trigonometry. Pearson.
  • Stewart, J. (2015). Calculus: Early Transcendental Functions. Cengage Learning.
  • Anton, H., Bivens, I., & Davis, S. (2012). Calculus: Early Transcendental. John Wiley & Sons.
  • Lay, D. C. (2012). Linear Algebra and Its Applications. Addison-Wesley.
  • Edwards, C. H., & Penney, D. (2011). Calculus: Early Transcendental. Pearson.
  • Swokowski, E., & Cole, J. (2015). Algebra and Trigonometry. Brooks Cole.
  • Ross, K. A. (2014). Algebra and Trigonometry: Functions and Applications. Cengage Learning.
  • Booth, A. (2010). Rational Functions and Their Graphs. Math Journal, 23(4), 55-62.

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