Im looking at internet sites and Im not quite understanding functions. I have a question like;

A function f(x) has the properties

i) f(1) = 1
ii) f(2x) = 4f(x)+6

Could you help me out. I know nothing about functions.

Differentiate both sides of:

f(2x) = 4f(x)+6 ----->

f'(2x) = 2f'(x) ------->

f'(x) = a x -------->

f(x) = b x^2 + c

Insert this in the functional equation to find that c = -2

The first equaton f(1) = 1 then gives:

f(x) = 3 x^2 - 2

thanks

oops wrong question

Of course! I'd be happy to explain how to approach this problem involving functions.

In mathematics, a function is a rule that maps each element from one set, called the domain, to exactly one element in another set, called the codomain. In simpler terms, a function takes an input value and produces an output value based on a specific rule. In this case, we have a function f(x).

Now, let's break down the problem step by step:

i) f(1) = 1
This equation tells us that when x is equal to 1, the function f returns a value of 1. In other words, plugging x = 1 into the function gives us f(1) = 1.

ii) f(2x) = 4f(x) + 6
This equation provides a relationship between the values of f(x) and f(2x). It says that when we have an input value of 2x, the output is equal to 4 times f(x) plus 6. This equation allows us to find the value of f(2x) given the value of f(x).

To solve this problem, we can use the given information to find the function value at different points. Let's start with a simple example: f(2).

According to property (ii), we can rewrite f(2) as f(2 * 1). Substituting x = 1 into the equation, we get:

f(2) = 4f(1) + 6

Since f(1) = 1 (given by property (i)), we rewrite the equation as:

f(2) = 4(1) + 6
f(2) = 4 + 6
f(2) = 10

So, we have determined that f(2) = 10.

By using similar logic, you can compute other values. For example, to find f(4), you can substitute x = 2 into equation (ii) and solve the resulting equation.

This approach allows you to find function values at specific points based on the given properties. If you need to find the function expression itself, additional information or methods might be required.

I hope this explanation helps you understand how to work with functions a little better. Feel free to ask any further questions you may have!