- Recognize that the composition \(g(f(x))\) is found by substituting the expression for \(f(x)\) into the variable \(x\) of \(g(x)\).
- Set up the substitution: \(g(f(x)) = (2x^3 + 3) - 4\).
- Simplify the expression: \(2x^3 + 3 - 4 = 2x^3 - 1\).
- The final expression is \(\boxed{2x^3 - 1}\).
Explanation
- Understanding the Composition of Functions
To find the value of the composite function \(g(f(x))\), we first examine the definitions of the functions \(f(x)\) and \(g(x)\). We are given \(f(x) = 2x^3 + 3\) and \(g(x) = x - 4\). The notation \(g(f(x))\) represents the composition of these two functions, which means we must evaluate \(g\) at the input \(f(x)\). This is achieved by substituting the expression for \(f(x)\) everywhere that \(x\) appears in the definition of \(g(x)\).
- Substituting and Simplifying the Expression
Given \(g(x) = x - 4\), substituting \(f(x)\) for \(x\) gives us:
\[g(f(x)) = f(x) - 4\]
Now, we replace \(f(x)\) with its specific definition, which is \(2x^3 + 3\):
\[g(f(x)) = (2x^3 + 3) - 4\]
By grouping like terms, we can simplify this expression:
\[g(f(x)) = 2x^3 + (3 - 4) = 2x^3 - 1\]
- Conclusion and Final Answer
Comparing our result \(2x^3 - 1\) with the provided options:
- \(2x^3 - 1\)
- \(2x^4 - 4x^3 + 3x - 12\)
- \(2x^3 - 24x^2 - 32x - 61\)
- \(2x^3 + 3x - 4\)
We clearly see that the correct option is the first one. Therefore, the value of \(g(f(x))\) is \(\boxed{2x^3 - 1}\).
Examples
Composite functions are essential in fields like computer science, where functions (or blocks of code) are frequently nested. For example, if a program has a function f(x) that calculates a discount and another function g(y) that calculates sales tax on a price y, the composite function g(f(x)) effectively calculates the final price after applying the discount first and then the tax. Understanding how to correctly substitute and simplify these expressions is fundamental for creating efficient and accurate software.