Question

Consider the following. f(x)= √x-9(x)=x+2 (a) Find the function (f. g)(x).

Answer 1

The function
`\((f \cdot g)(x)\) is \(f(x) \cdot g(x) = (√(x-9))(x+2)\).`

Step-by-step explanation:

To find the product of two functions,
`\(f(x)\) and \(g(x)\)` , denoted as
`\((f \cdot g)(x)\)` , we multiply the expressions for
`\(f(x)\) and \(g(x)\)` . For the given functions
`\(f(x) = √(x-9)\)` and
`\(g(x) = x+2\),` the product is expressed as
`\((f \cdot g)(x) = (√(x-9))(x+2)\).`

Expanding the expression yields
`\((f \cdot g)(x) = √(x-9) \cdot x + √(x-9) \cdot 2\).` This can be further simplified to
`\(x√(x-9) + 2√(x-9)\).`

Understanding how to find the product of two functions is essential in calculus, as it allows us to combine the effects of different functions on a given variable. In this case, the product
`\((f \cdot g)(x)\)` represents the combined effect of
`\(f(x)\)` and
`\(g(x)\)` on the variable
`\(x\).`

Knowing how to manipulate and simplify expressions involving square roots and variables enables a deeper understanding of mathematical functions and their interactions. The product of functions is a fundamental concept in calculus and has applications in various mathematical models and real-world scenarios.