To find the product function $$(f \cdot g)(x)$$ for the polynomial functions $$f(x) = x + 2$$ and $$g(x) = x^2 - 3x - 4$$, substitute the expressions into the multiplication formula: $$(f \cdot g)(x) = (x + 2)(x^2 - 3x - 4)$$. Multiply the polynomials by distributing: $$x(x^2 - 3x - 4) + 2(x^2 - 3x - 4)$$. This expands to $$x^3 - 3x^2 - 4x + 2x^2 - 6x - 8$$. Combining the like terms yields the final product function: $$(f \cdot g)(x) = x^3 - x^2 - 10x - 8$$.

Multiplying $$f(x) = x + 2$$ and $$g(x) = x^2 - 3x - 4$$

To find the product function $$(f \cdot g)(x)$$ for $$f(x) = x - 5$$ and $$g(x) = x^2 - 2x + 3$$, substitute the expressions into the formula $$(f \cdot g)(x) = f(x) \cdot g(x)$$. Multiplying the polynomials gives $$(x - 5)(x^2 - 2x + 3)$$. Distribute the terms to obtain $$x^3 - 2x^2 + 3x - 5x^2 + 10x - 15$$. Combining like terms yields the simplified product function: $$(f \cdot g)(x) = x^3 - 7x^2 + 13x - 15$$.

Multiplying $$f(x) = x - 5$$ and $$g(x) = x^2 - 2x + 3$$

To find the product function $$(f \cdot g)(x)$$ for $$f(x) = x - 7$$ and $$g(x) = x^2 + 8x + 4$$, use the formula $$(f \cdot g)(x) = f(x) \cdot g(x)$$ and multiply the polynomials: $$(x - 7)(x^2 + 8x + 4)$$. Distributing the terms produces $$x^3 + 8x^2 + 4x - 7x^2 - 56x - 28$$. Combining like terms simplifies the product to $$(f \cdot g)(x) = x^3 + x^2 - 52x - 28$$.

Multiplying $$f(x) = x - 7$$ and $$g(x) = x^2 + 8x + 4$$

For polynomial functions $$f(x)$$ and $$g(x)$$, their multiplication is defined by the formula $$(f \cdot g)(x) = f(x) \cdot g(x)$$. This means that to find the product of two functions, you multiply their respective polynomial expressions together.

Claude

In mathematics, equations that define functions are often written using function notation. Functions are typically assigned single-letter names, most commonly $$f$$, $$g$$, $$h$$, $$F$$, $$G$$, or $$H$$. For a function named $$f$$, the notation specifies the relationship between an input value $$x$$ and its corresponding output value $$y$$ as $$y = f(x)$$. In this notation, $$x$$ represents the domain value, and $$f(x)$$ represents the corresponding range value. The expression $$f(x)$$ is read as '$$f$$ of $$x$$' or 'the value of $$f$$ at $$x$$'. The parentheses in this context indicate the input parameter mapping to the function and do not indicate multiplication.

Function Notation

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For a mathematical function expressed in the notation $$y = f(x)$$, the input variable $$x$$ is known as the independent variable. It is referred to as 'independent' because it can be assigned any valid value from the domain of the function.

Independent Variable of a Mathematical Function

For a mathematical function expressed in the notation $$y = f(x)$$, the output variable $$y$$ (which is equivalently written as $$f(x)$$) is known as the dependent variable. It is referred to as 'dependent' because its specific value relies entirely on the value chosen for the independent variable $$x$$.

Dependent Variable of a Mathematical Function

Evaluating a mathematical function is the process of finding the numerical value of $$f(x)$$ for a specific given value of the independent variable $$x$$. To perform this process, substitute the provided value for $$x$$ into the function's equation, and then simplify the resulting numerical expression by following the order of operations.

Evaluating a Mathematical Function

The graph of a mathematical function is the visual representation of all its ordered pairs $$(x, y)$$ on a coordinate system. Because a function can be written using function notation where $$y = f(x)$$, these ordered pairs can equivalently be expressed as $$(x, f(x))$$. In this notation, $$f$$ represents the name of the function, $$x$$ is the $$x$$-coordinate of the ordered pair (the input), and $$f(x)$$ is the corresponding $$y$$-coordinate (the output). Thus, graphing a relation like $$f(x) = 2x - 3$$ produces the exact same line as graphing $$y = 2x - 3$$; only the notation has changed.

Graph of a Mathematical Function

Multiplication of Polynomial Functions

For polynomial functions $$f(x)$$ and $$g(x)$$, where $$g(x) 
eq 0$$, their division is defined by the formula $$(f/g)(x) = \frac{f(x)}{g(x)}$$. This means that to find the quotient function, you set up a rational expression with the polynomial for $$f(x)$$ in the numerator and the polynomial for $$g(x)$$ in the denominator. The resulting expression can then be simplified using polynomial division.

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