Operations on Functions

Question

If f(x) = x²-1, then what is f(a+b)? What is f(a)-f(b)? Are f(a*b) and f(a)*f(b) the same? What is f(f(a))?

Answer 1

f(x) = x² - 1

f(a+b) = (a+b)² - 1 = a² + 2ab + b² - 1

f(a) - f(b) = (a² - 1) - (b² - 1) = a² - b²

f(f(a) = f(a² - 1) = (a²-1)² - 1 = a⁴ - 2a² + 1 - 1 = a⁴ - 2a²

Answer 2

Hello, thank you for taking the time to post your question!

If you’re starting with f(x) = x^2 – 1, it’s just a matter of making the substitutions and then simplifying. So for this set of expressions that would mean taking

f(a + b) = (a + b)^2 - 1 = a^2 + 2ab + b^2 - 1

f(a) – f(b) = a^2 – 1 – (b^2 – 1) = a^2 – b^2

f(a*b) = (ab)^2 – 1 = a^2b^2 - 1

f(a) * f(b) = (a^2 – 1)(b^2 – 1) = a^2b^2 – a^2 – b^2 + 1

f(f(a)) = f(a^2 – 1) = (a^2 – 1)^2 – 1 = a^4 – 2a^2

Hopefully that gets you moving in the right direction! Feel free to reach out with a message if you have any questions beyond that! :)

Answer 3

f(x) = x^2 -1

f(a+b) = (a+b)^2 -1 = a^2 +2ab +b^2 - 1

f(a) - f(b) = a^2 -1 - (b^2 -1) = a^2 -b^2

f(f(a)) = f(a^2 -1) = (a^2-1)^2 -1 = a^4 -2a^2 +1 -1 = a^4 -2a^2

Answer 4

Let's start with the first question: f(a+b). Here we substitute (a+b) into the x of the original question and get (a+b)²-1. If we want to plug in numbers, say a=3 and b=2, then we get (3+2)²-1 = 5²-1 = 25-1 = 24.

The second questions asks f(a) - f(b). Notice how the subtraction operation is outside of the function this time. This means we should calculate f(a) and f(b) separately before finding the difference between the two. Let's use the same example where a = 3 and b = 2. f(a) = f(3) = 3²-1 = 9-1 = 8. f(b) = f(2) = 2²-1 = 4-1 = 3. Now we do the subtraction f(a) - f(b) = 8 - 3 = 5.

The third question asks if there is a difference between f(a*b) versus f(a)*f(b). Let's take a look. f(a*b) = (a*b)²-1 = a²b²-1 while f(a)*f(b) = (a²-1)*(b²-1) = a²b²-a²-b²+1. These are two different expressions, so the answer is that these will not always be the same.

Now let's look at our last question. Here we are taking a function of a function. How we approach this is by first calculating the inner expression (f(a)) and plugging the result into the function. f(a) = a²-1. Now we take the result and put it into the function to get our answer which can be written f(f(a)) = f(a²-1) = (a²-1)²-1. If we let a = 3, then we get f(f(3)) = (3²-1)²-1 = (9-1)²-1 = 8²-1 = 64-1 = 63.