How Does the Function f(1/x) Simplify to the Book's Answer?

[CarlosRamos]

I'm having difficulties in obtaining the book/calculator answer to the function
f(x) = (x-1)(x+5) when f(1/x).

The book answer is: [(1-a)(1+5a)]/a
The answer I get is: 1/(a^2) + 4a - 5
. or : (1 + 4a - 5a^2)/a^2
. or : [(1 + 4a)/a^2] - 5

I know that all these answers are possible answers, but I can't for the life of me figure out how the book got the answer (I've also checked with a graphing calculator, and received the same answers)

- Thanks :)

 

[courtrigrad]

$$f(x) = (x-1)(x+5)$$

Then $$f(\frac{1}{a}) = (\frac{1}{a}-1)(\frac{1}{a} +5 )$$

Try working with the fractions (i.e. adding and subtracting) within the parentheses, then multiply them and see what you get.

Also it shoud be $$\frac{(1-a)(1+5a)}{a^{2}}$$

 

[Architeuthis Dux]

I understand your frustration in trying to obtain the answer to this function. It is important to remember that there are multiple ways to represent and simplify a function, and different methods may yield different answers. In this case, it seems that the book is using a specific method that is different from the method you are using.

To obtain the book's answer, it may be helpful to expand the function f(x) = (x-1)(x+5) when f(1/x) by substituting 1/x for x. This will result in [(1/x)-1][(1/x)+5]. From here, we can simplify by multiplying the terms together and combining like terms to get [(1-1/x)(1+5/x)].

Next, we can use the rule of simplifying fractions by multiplying the numerator and denominator by the same number to get an equivalent fraction. In this case, we can multiply the numerator and denominator by x to get [(x-1)(x+5)]/x.

Finally, we can substitute a for 1/x to get [(1-a)(1+5a)]/a, which is the book's answer.

I hope this explanation helps you understand how the book obtained their answer. Remember, there are often multiple ways to simplify a function, so it is important to understand the method being used in order to obtain the correct answer. Keep practicing and don't get discouraged, as understanding and solving complex functions takes time and practice.