Question

If \(\mathrm{f}(\mathrm{x})=2 \mathrm{x}\) and \(\mathrm{g}\) is identity function, then (a) \((\operatorname{fog})(\mathrm{x})=\mathrm{g}(\mathrm{x})\) (b) \((\operatorname{gog})(x)=g(x)\) (c) \((\) fog \()(x)=(g+g)(x)\) (d) \((\mathrm{fog})(\mathrm{x})=(\mathrm{f}+\mathrm{f})(\mathrm{x})\)

Short answer

Statements (b) and (c) are true: (b) \((gog)(x) = g(x)\), and (c) \((fog)(x) = (g+g)(x)\). Statements (a) and (d) are false.

Step-by-step solution

Compute the Composition of f and g: (fog)(x)

To find the composition of functions f and g, we need to replace the variable x in the f function with the entire g function. This is represented as (fog)(x) or f(g(x)): \((fog)(x) = f(g(x))\) Now, substitute the given functions: \((fog)(x) = f(x) = 2x\)

Check if (a) is True

To check if statement (a) is true, we need to compare (fog)(x) with g(x). We have already computed: \((fog)(x) = 2x\) and we know, \(g(x) = x\) Since \((fog)(x) \neq g(x)\), statement (a) is false.

Compute the Composition of g and g: (gog)(x)

To find the composition of function g with itself, we need to replace the variable x in the g function with the entire g function. This is represented as (gog)(x) or g(g(x)): \((gog)(x) = g(g(x))\) Now, substitute the given functions: \((gog)(x) = g(x) = x\)

Check if (b) is True

To check if statement (b) is true, we need to compare (gog)(x) with g(x). We have already computed: \((gog)(x) = x\) and we know, \(g(x) = x\) Since \((gog)(x) = g(x)\), statement (b) is true.

Check if (c) is True

Statement (c) states that \((fog)(x) = (g+g)(x)\) We already know that: \((fog)(x) = 2x\) and we need to compute: \((g+g)(x) = g(x) + g(x) = x + x = 2x\) Since \((fog)(x) = (g+g)(x)\), statement (c) is true.

Check if (d) is True

Statement (d) states that \((fog)(x) = (f+f)(x)\) We already know that: \((fog)(x) = 2x\) and we need to compute: \((f+f)(x) = f(x) + f(x) = 2x + 2x = 4x\) Since \((fog)(x) \neq (f+f)(x)\), statement (d) is false. In conclusion, statements (b) and (c) are true, while statements (a) and (d) are false.

Key concepts

Function Composition

Function composition is a process where two functions are combined to form a new function, where the output of one function becomes the input of the second. This can feel a bit like a relay race, where one function 'passes the baton' to the next. In mathematical terms, if we have two functions, say, f and g, the composition (fog) is read as 'f composed with g' and is written as f(g(x)). The gist here is that the function g is applied first, and then the function f is applied to the result of g.

In the provided exercise, f(x) is defined as 2x, and g is the identity function, which basically means that g(x) is x - it does nothing but return the input as is. Understanding this, we can solve various parts of the given exercise by correctly composing f and g.

Let's illustrate this with a simple real-life example: Imagine you have an app that doubles the number of likes on a post, and another app that simply shows the current number of likes. If you apply the second app to a post first (the identity function, g(x)) and then the doubling app (f(x)), you would get double the original likes, showcasing how function composition works in practice.

Identity Function

The identity function is somewhat of a 'do nothing' function. It's like asking someone to change something, and they just hand you back the same thing, unchanged. Mathematically, the identity function g(x) simply gives back the input value of x. In symbols, it's written as g(x) = x. This function plays a key role in understanding the nature of function composition because it's essentially the 'neutral element' of this operation.

Using the analogy of a color-changing machine, where you put in an object and it changes the color based on a function, the identity function would be like a machine that you put the object into and it comes out unchanged. This concept ensures that when we compose the identity function with any other function, the original function's properties remain intact. As seen in part (b) of the provided exercise, when composing g with itself (gog)(x), we essentially get g(x) back, which proves that (gog)(x) = x, thus highlighting the identity function's unique property.

Mathematical Operations

Mathematical operations are the bread and butter of algebra—addition, subtraction, multiplication, and division. When we deal with functions, these operations take on a new life as they can be combined and used to manipulate functions in various ways.

In the context of the exercise, we observe operations in action in statement (c), where the sum of two functions, (g+g)(x), is considered. Here, we're not adding numbers, but rather the outputs of the identity function when applied to x. With g(x) = x, the operation becomes x + x, which simplifies to 2x. This mirrors the composition of (fog)(x).

But not all operations yield the same results with functions. For example, in statement (d), we see that (f+f)(x) is not the same as (fog)(x) because f(x) + f(x) is 2x + 2x, which is 4x, not 2x. This shows that function composition doesn't necessarily follow the same rules as basic arithmetic with numbers and highlights the importance of understanding each mathematical operation within the context of functions.