Two functions can be expressed as the sum of two other functions, which is called composition. Composition is when one function (the noun) is expressed as the output of another function (the verb) and that result is input to a third function.
For example, if M(x) = X2 + 3 and N(x) = 5x + 9, then M(x) can be expressed as the sum of the following two functions:
1. The function Y(x) = x + 5, where x is the input and y is the output.
2. The function (Mn)(x), where n is any integer. (Mn)(x) = xn.
This article will explain how to find (Mn)(x).
M(x) = 5M(x/5) + 9
In this bullet point, we discuss how to find an expression that is equivalent to (Mn)(x) where Mn is a different expression with n variables.
Examine the coefficients of each variable in the given equation. The coefficient of x is nothing, so this variable can be eliminated. The coefficient of n is 1, so this variable cannot be eliminated.
Now, determine which variables can be combined. Can any pairs of variables be combined into one variable? If so, do so! Can any variables be multiplied or divided? If so, do so!
Once you have done that, assemble the remaining variables into a new expression using the same rules and combine it with n = 1. Then solve for the new variable using algebraic methods.
N(x) = (Mn)(x)
The word “equivalent” in the question asks you to show that two expressions are the same. When you work with algebra, you will often encounter situations where you need to find the variable expression that represents a given equation.
The equation N(x) = (Mn)(x) asks you to find the variable expression that represents the expression M(x)n.
To do this, you need to prove that every element in M(x)n is represented in N(x). You will also need to show that each element in M(x)n is in the correct order.
Let’s look at some examples to make this clearer. Suppose we have two expressions: (A + B)(1 + 2) and (C + D)(3 + 4). We want to find equivalent expressions for these.
N(x) = 5N(x/5)+ 9
In this case, M(x) = N(x), so you would be solving for N(x) in the above equation.
Using the same method as above, you would get N(x) = 5N(x/5)+ 9, or N(x) = (5 x)/5 + 9. Solving for x gives you the answer: x = 9/5.
Notice that this answer is in the form of a fraction, which makes sense since 5N is a fraction! Using algebra to combine numbers can sometimes yield unexpected results like this.
Algebra can be used in many different situations, so make sure to always check if your answer is in the correct form before proceeding with your problem.
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