What is the composition of the functions f(x) = 3x^2 and g(x) = 5x + 10?

To find g(x) ∘ f(x) or g(f(x)), we start with the definitions of the two parent functions.

Given:

• f(x) = 3x²
• g(x) = 5x + 10

Now, we will substitute f(x) into g(x). This means that wherever we have ‘x’ in g(x), we will replace it with f(x).

Therefore:

g(f(x)) = g(3x²)

Substituting into the equation:

g(f(x)) = 5(3x²) + 10

This simplifies as follows:

g(f(x)) = 15x² + 10

So, the composition of the functions results in:

g(f(x)) = 15x² + 10

This composition represents a new function that shows how the outputs of f(x) influence the outputs of g(x).

It’s also interesting to note that the original functions f(x) and g(x) have specific behaviors: f(x) is a quadratic function which opens upwards, and g(x) is a linear function. The composition, g(f(x)), results in another quadratic function, demonstrating the interplay between linear and quadratic functions.

This function will also have its own properties in terms of vertex, intercepts, and its general shape on the graph.