What is the product of the parent functions f(x) = log10(x) and g(x) = 5x²?

Finding the Product of f(x) and g(x)

To find the product of the two given functions, we first need to define them clearly:

• f(x) = log₁₀(x)
• g(x) = 5x²

The product of these two functions, denoted as f(x) * g(x), can be expressed mathematically as:

f(x) * g(x) = log₁₀(x) * 5x²

Now, we can simplify this to:

f(x) * g(x) = 5x² * log₁₀(x)

This expression represents the product of the two functions. To analyze its behavior:

• Domain: The function f(x) = log₁₀(x) is defined for x > 0. Therefore, the product f(x) * g(x) is also defined for x > 0.
• Range: Since g(x) = 5x² is always non-negative for real values of x, the product will take on a range that depends on the behavior of log₁₀(x).

As x approaches 0 from the positive side, log₁₀(x) approaches negative infinity. Conversely, as x increases, log₁₀(x) increases without bound. Therefore, the product will exhibit interesting behavior near these extremes.

In summary, the product of the parent functions is:

f(x) * g(x) = 5x² * log₁₀(x)

This expression combines logarithmic and polynomial features, which can be useful in various applications in calculus and analysis.