Consider the functions f , g : R → R defined as f (x) = 3px + 1 and g(x) = x3. Find theformulas for g ◦ f and f ◦ g

To find the formulas for g ◦ f and f ◦ g, we need to substitute the function g into f and vice versa.

First, let's find the formula for g ◦ f. To do this, we substitute f(x) into g(x):

g ◦ f(x) = g(f(x))

Substituting f(x) = 3px + 1 into g(x), we get:

g ◦ f(x) = (3px + 1)^3

Expanding the expression, we have:

g ◦ f(x) = (3px + 1)(3px + 1)(3px + 1)

Simplifying further, we get:

g ◦ f(x) = (9p^2x^2 + 6px + 1)(3px + 1)

Expanding again, we have:

g ◦ f(x) = 27p^3x^3 + 18p^2x^2 + 3px + 9p^2x^2 + 6px + x

Combining like terms, we get the final formula for g ◦ f:

g ◦ f(x) = 27p^3x^3 + 27p^2x^2 + 9px + x

Now, let's find the formula for f ◦ g. To do this, we substitute g(x) into f(x):

f ◦ g(x) = f(g(x))

Substituting g(x) = x^3 into f(x), we get:

f ◦ g(x) = 3p(x^3) + 1

Simplifying further, we have:

f ◦ g(x) = 3px^3 + 1

Therefore, the formulas for g ◦ f and f ◦ g are:

g ◦ f(x) = 27p^3x^3 + 27p^2x^2 + 9px + x

f ◦ g(x) = 3px^3 + 1