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: f(x) = x^2 + 11

Step-by-step solution

Understand the composition

The composition \((f \circ g)(x)\) is defined as \(f(g(x))\). Our goal is to find the function \(f(x)\) such that when \(g(x) = x^2 + 3\) is substituted into it, we get the given composition \(x^4 + 6x^2 + 20\).

Substitute g(x) into the composition

Let's substitute \(g(x)\) into the composition. Since \(g(x) = x^2 + 3\), we can write \((f \circ g)(x)\) as: $$ f(x^2 + 3) $$

Express the given composition in terms of f(x)

Now, we are given that \((f \circ g)(x) = x^4 + 6x^2 + 20\). This means that: $$ f(x^2 + 3) = x^4 + 6x^2 + 20 $$

Identify the form of f(x)

We can see that the expression on the right side of the equation is a quadratic equation in terms of \((x^2 + 3)\). This suggests that the function \(f(x)\) should also be a quadratic function. Let us assume that: $$ f(x) = ax^2 + bx + c $$

Substitute g(x) into f(x)

Now, let's substitute \(g(x) = x^2 + 3\) into \(f(x)\): $$ f(x^2 + 3) = a(x^2 + 3)^2 + b(x^2 + 3) + c $$

Expand and simplify the expression

Expand and simplify the expression on the right side of the equation: $$ a(x^4 + 6x^2 + 9) + b(x^2 + 3) + c $$ $$ ax^4 + 6ax^2 + 9a + bx^2 + 3b + c $$ $$ ax^4 + (6a + b)x^2 + (9a + 3b + c) $$

Match the coefficients

Now, we need to match the coefficients of the given composition \(x^4 + 6x^2 + 20\) with the expression derived from our substitution: $$ x^4 + 6x^2 + 20 = ax^4 + (6a + b)x^2 + (9a + 3b + c) $$ From this, we can equate the coefficients of the corresponding terms: 1. \(a = 1\) 2. \(6a + b = 6\) 3. \(9a + 3b + c = 20\)

Solve the system of equations

Now, let's solve the system of equations: 1. \(a = 1\) 2. \(6(1) + b = 6 \Rightarrow b = 0\) 3. \(9(1) + 3(0) + c = 20 \Rightarrow c = 11\)

Write the function f(x)

Now that we have the values of \(a\), \(b\), and \(c\), we can write the function \(f(x)\): $$ f(x) = 1x^2 + 0x + 11 $$ $$ f(x) = x^2 + 11 $$ This is the function \(f(x)\) that produces the given composition \((f \circ g)(x) = x^4 + 6x^2 + 20\) when composed with the function \(g(x) = x^2 + 3\).