Question

Let \(g(x)=x^{2}+3 .\) Find a function \(f\) that produces the given composition. $$(f \circ g)(x)=x^{4}+6 x^{2}+20$$

Short answer

Answer: The function \(f(x)\) that produces the given composition is \(f(x) = x^2 + 6x^2 + 17\).

Step-by-step solution

Identify the known function and the given composition

We are given the function \(g(x) = x^2 + 3\) and the composition \((f \circ g)(x) = x^4 + 6x^2 + 20\).

Substitute the known function into the composition

Replace the \(x\) in \((f \circ g)(x)\) with the expression for \(g(x)\). That is, let \(x = g(x) = x^2 + 3\). So the composition becomes: $$ (f \circ g)(x^2 + 3) = (x^2 + 3)^4 + 6(x^2 + 3)^2 + 20. $$

Simplify the expression obtained in Step 2

Expand the terms in the expression: $$ (x^2 + 3)^4 + 6(x^2 + 3)^2 + 20 = x^4 + 6x^2 + 20. $$ This expression represents the value of \((f \circ g)(x)\) after substituting \(g(x)\).

Identify the function f

From the expression above, we can identify the function \(f\). Notice that the expression is the same format as \(g(x) = x^2 + 3\), but with different constants. Therefore, we can write the function \(f\) as: $$ f(x) = x^2 + 6x^2 + 20 - 3 = x^2 + 6x^2 + 17. $$

Conclusion

The function \(f(x)\) that produces the given composition is $$ f(x) = x^2 + 6x^2 + 17. $$