Introduction to Inverse Functions

In calculus and algebra, understanding the concept of inverse functions is essential. An inverse function essentially reverses the operation of another function. This means that if you apply a function and then its inverse, you return to your original input.

Step-by-Step Guide to Finding the Inverse Function

Let’s explore the inverse function of fx -4x - 5. This will be our primary focus, and we will also tackle similar examples to solidify our understanding.

Example 1:

Find the inverse of fx -4x - 5.

Replace fx with y: t

y -4x - 5

Swap x and y: t

x -4y - 5

Solve for y: t ttFirst, add 5 to both sides: tt tttx 5 -4y tt ttThen divide by -4: tt ttty -frac{x 5}{4} tt t Express the inverse function: t

f-1x -frac{x 5}{4}

tConclusion: The inverse function of fx -4x - 5 is f-1x -frac{x 5}{4}.

Additional Examples

Example 2:

tfx 4 - x tLet y fx t tty 4 - x t t tx 4 - y ty fx then f-1y x t ttf-1y 4 - y t t tReplace y by x t ttf-1x 4 - x t t tConclusion: The inverse of fx 4 - x is f-1x 4 - x.

Example 3:

tfx 4 - x tSwapping x and y yields: x 4 - y tFind y in terms of x we get: y 4 - x tConclusion: fx 4 - x is its own inverse.

Example 4:

tx g(x) f-1x tx f(g(x)) g(x) - frac{4}{g(x) - 5} tx(g(x) - 5) g(x) - 4 txg(x) - 5x g(x) - 4 txg(x) - g(x) 5x - 4 tg(x)(x - 1) 5x - 4 tf-1x g(x) frac{5x - 4}{x - 1} tConclusion: The inverse function of g(x) is f-1x frac{5x - 4}{x - 1}.

Example 5:

tfx frac{x - 4}{x - 5} tLet y frac{x - 4}{x - 5} tBy interchanging x and y: t ttx frac{y - 4}{y - 5} t t tMultiply both sides by xy - 5x y - 4 tRearrange to isolate y: t tty - xy 4 - 5x t t tFactor out y: t tty(1 - x) 4 - 5x t t tDivide by 1 - x to solve for y: t tty frac{4 - 5x}{1 - x} t t tReplace y by f-1x t ttf-1x frac{4 - 5x}{1 - x} t t tConclusion: The inverse function of fx frac{x - 4}{x - 5} is f-1x frac{4 - 5x}{1 - x}.

Example 6:

tfx 4x - 5 to find the inverse, first interchange x and y: tx 4y - 5 tSolve for y: t ttAdd 5 to both sides: tt tttx 5 4y tt ttDivide both sides by 4: tt ttty frac{x 5}{4} tt t t tThis is the inverse: t ttfx 4x - 5 has an inverse of f-1x frac{x 5}{4}. t tConclusion: The inverse function of fx 4x - 5 is f-1x frac{x 5}{4}.

Conclusion

Understanding the concept and the process of finding the inverse of a function is crucial for advanced mathematics and practical applications. Whether you are dealing with linear functions, rational functions, or more complex equations, the steps to find the inverse follow a systematic approach. Practice is key, and mastering these steps will enhance your problem-solving skills in algebra and calculus.