- Calculate the inner function value: \(f(2) = 4(2)^3 - 10 = 22\).
- Use the result of the inner function as input for the outer function: \(g(22) = \frac{3(22) - 4}{2}\).
- Evaluate the arithmetic: \(\frac{66 - 4}{2} = 31\).
- The final value is \(\boxed{31}\).
Explanation
- Analysis of the Problem
We are given two functions, \(f(x) = 4x^3 - 10\) and \(g(x) = \frac{3x - 4}{2}\). Our objective is to find the composite function value \(g(f(2))\). This involves a two-step process: first, calculating the output of the inner function \(f(x)\) when \(x=2\), and then using that output as the input for the outer function \(g(x)\).
- Evaluating the Inner Function \(f(2)\)
To evaluate \(f(2)\), we substitute \(x=2\) into the expression for \(f(x)\):
\[ f(2) = 4(2)^3 - 10 \]
First, evaluate the power: \(2^3 = 8\).
Then, perform the multiplication: \(4 \times 8 = 32\).
Finally, subtract 10: \(32 - 10 = 22\).
Thus, the output of the inner function is \(f(2) = 22\).
- Evaluating the Composite Function \(g(f(2))\)
Now we substitute the result from the previous step, \(f(2) = 22\), into the function \(g(x)\):
\[ g(f(2)) = g(22) = \frac{3(22) - 4}{2} \]
First, evaluate the numerator: \(3 \times 22 = 66\).
Then, subtract 4 from the result: \(66 - 4 = 62\).
Finally, divide by 2: \(\frac{62}{2} = 31\).
Therefore, the value of the composite function \(g(f(2))\) is 31.
Examples
Function composition is like a factory assembly line. Suppose a raw material (the input \(x=2\)) is processed by one machine (\(f(x)\)) which adds or modifies it according to its function, and the output of that machine serves as the input for a second machine (\(g(x)\)). This concept is used everywhere in computer programming, where the output of one function becomes the input of another, or in finance, where the result of one tax calculation is then used to determine a final net payout. Understanding this allows you to build complex processes by chaining simpler, well-defined steps.