- Calculate \((f \circ g)(x) = f(g(x)) = 4(x^2 - 3)^2 + 1 = 4(x^4 - 6x^2 + 9) + 1 = 4x^4 - 24x^2 + 37\).
- Calculate \((g \circ f)(x) = g(f(x)) = (4x^2 + 1)^2 - 3 = 16x^4 + 8x^2 + 1 - 3 = 16x^4 + 8x^2 - 2\).
- The compositions are:
\((f \circ g)(x) = \boxed{4x^4 - 24x^2 + 37}\)
\((g \circ f)(x) = \boxed{16x^4 + 8x^2 - 2}\)
Explanation
- Understanding Function Composition
To solve the compositions of the functions \(f(x) = 4x^2 + 1\) and \(g(x) = x^2 - 3\), we need to substitute one function into the other. Function composition \((f \circ g)(x)\) is defined as \(f(g(x))\), and \((g \circ f)(x)\) is defined as \(g(f(x))\). We will proceed by substituting the expressions and expanding the resulting polynomials.
- Calculating \((f \circ g)(x)\)
To find \((f \circ g)(x)\), we substitute \(g(x) = x^2 - 3\) into the function \(f(x) = 4x^2 + 1\):
\((f \circ g)(x) = f(x^2 - 3) = 4(x^2 - 3)^2 + 1\).
Expanding the squared term:
\((x^2 - 3)^2 = x^4 - 6x^2 + 9\).
Now, multiply by 4 and add 1:
\(4(x^4 - 6x^2 + 9) + 1 = 4x^4 - 24x^2 + 36 + 1 = 4x^4 - 24x^2 + 37\).
Thus, \((f \circ g)(x) = 4x^4 - 24x^2 + 37\).
- Calculating \((g \circ f)(x)\)
To find \((g \circ f)(x)\), we substitute \(f(x) = 4x^2 + 1\) into the function \(g(x) = x^2 - 3\):
\((g \circ f)(x) = g(4x^2 + 1) = (4x^2 + 1)^2 - 3\).
Expanding the squared term:
\((4x^2 + 1)^2 = (4x^2)^2 + 2(4x^2)(1) + 1^2 = 16x^4 + 8x^2 + 1\).
Now, subtract 3:
\((16x^4 + 8x^2 + 1) - 3 = 16x^4 + 8x^2 - 2\).
Thus, \((g \circ f)(x) = 16x^4 + 8x^2 - 2\).
- Conclusion
Based on our calculations:
\((f \circ g)(x) = 4x^4 - 24x^2 + 37\)
\((g \circ f)(x) = 16x^4 + 8x^2 - 2\)
These expressions match two of the options provided: \(4x^4 - 24x^2 + 37\) and \(16x^4 + 8x^2 - 2\). The final answers are \(\boxed{4x^4 - 24x^2 + 37}\) and \(\boxed{16x^4 + 8x^2 - 2}\).
Examples
Function composition is widely used in computer science for piping the output of one function directly into the input of another. For example, if you have a function that converts currency (e.g., USD to EUR) and another that calculates the tax on a price in EUR, composing them allows you to find the total cost in tax-adjusted EUR directly from a USD input. Mathematically, if \(C(u)\) is the currency conversion and \(T(e)\) is the tax calculation, the total process is represented by the composed function \((T \circ C)(u)\). This abstraction simplifies complex processes into clear, manageable steps.