To describe: The difference between (fg)(x) and (f⋅g)(x)

Question

Corben Pittman · Accepted Answer

Formula used:
Product of two functions:

(f g) (x) = f (x) g (x)

Composition of two functions:

(f \cdot g) (x) = f (g (x))

Calculation:
The function (fg)(x) denotes the product of two functions f and g and can be written as

(f g) (x) = f (x) g (x)

.
The function

(f \cdot g) (x)

denotes the composition of two functions f and g and can be written as

(f \cdot g) (x) = f (g (x))

.
Let us see their examples. Suppose two functions are:

f (x) = 3 x^{2}, g (x) = x + 1

Therefore,

(f g) (x) = f (x) g (x) = 3 x^{2} (x + 1) = 3 x^{3} + 3 x^{2}

(f \cdot g) (x) = f (g (x)) = 3 {(g (x))}^{2} = 3 {(x + 1)}^{2} = 3 (x^{2} + 1 + 2 x) = 3 x^{2} + 6 x + 3

Conclusion:
The function (fg)(x) denotes the product of two functions f and g and can be written as

(f g) (x) = f (x) g (x)

.
The function

(f \cdot g) (x)

denotes the composition of two functions f and g and can be written as

(f \cdot g) (x) = f (g (x))

.

Accepted Answer

(g°f) (x)

RizerMix · Accepted Answer

Step 1: Given:(fg)(x) represents the composition of two functions, f and g. It means that we first apply the function g to the input x, and then we apply the function f to the result. Mathematically, (fg)(x) can be written as f(g(x)).On the other hand, (f*g)(x) represents the pointwise multiplication or product of two functions, f and g. It means that for each value of x, we multiply the outputs of f(x) and g(x). Mathematically, (f*g)(x) can be written as f(x)·g(x).Step 2: Finally, the difference between (fg)(x) and (f*g)(x) lies in the operations performed. (fg)(x) represents the composition of two functions, where we apply g first and then f, while (f*g)(x) represents the pointwise multiplication of two functions, where we multiply the outputs of f(x) and g(x) for each input value of x.Here&#039;s a visual representation of the difference:For (fg)(x):1. Apply g to x: g(x)2. Apply f to the result: f(g(x))For (f*g)(x):1. Evaluate f(x): f(x)2. Evaluate g(x): g(x)3. Multiply the results: f(x)·g(x)

Don Sumner · Accepted Answer

To describe the difference between (fg)(x) and (f*g)(x):(fg)(x) represents the composition of two functions f and g, where the output of g is used as the input for f. In mathematical notation, it can be written as (fg)(x)=f(g(x)).On the other hand, (f*g)(x) represents the product of two functions f and g, where the output of f is multiplied by the output of g. In mathematical notation, it can be written as (f*g)(x)=f(x)·g(x).In summary, (fg)(x) is the composition of functions, while (f*g)(x) is the product of functions.

nick1337 · Accepted Answer

Answer:- (fg)(x) is the product of f(x) and g(x): (fg)(x)=f(x)·g(x).- (f*g)(x) is the convolution of f(x) and g(x): (f*g)(x)=∫−∞∞f(t)·g(x−t)dt.Explanation:In mathematics, (fg)(x) represents the product of two functions, f(x) and g(x). It is obtained by multiplying the outputs of f(x) and g(x) evaluated at the same input, x. Mathematically, it can be expressed as (fg)(x)=f(x)·g(x).On the other hand, (f*g)(x) represents the convolution of two functions, f(x) and g(x). Convolution is an operation that combines two functions to produce a third function, which describes how the shape of one function affects the other as it is shifted and scaled. Mathematically, the convolution (f*g)(x) is defined as:(f*g)(x)=∫−∞∞f(t)·g(x−t)dtIn simpler terms, to evaluate (f*g)(x), we take one function, f(x), and &#039;&#039;slide&#039;&#039; it across the other function, g(x), while multiplying their values at each point and adding up the results.Therefore, the main difference between (fg)(x) and (f*g)(x) lies in the operation performed. (fg)(x) represents a point-wise multiplication of two functions, while (f*g)(x) represents their convolution.

To describe: The difference between (fg)(x) and (f \cdot g)(x)

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