- Identify the functions \(f(x) = -3x + 3\) and \(g(x) = 2x^2 + 4x - 8\).
- Calculate the value of the inner function at \(x=1\), which gives \(f(1) = -3(1) + 3 = 0\).
- Substitute the result into the outer function \(g(f(1)) = g(0) = 2(0)^2 + 4(0) - 8 = -8\).
- The final value of the composition is \(\boxed{-8}\).
Explanation
- Problem Analysis
The problem requires us to evaluate the composition of two functions, \(f(x)\) and \(g(x)\), at a specific point. We are given the functions:
\[f(x) = -3x + 3\]
\[g(x) = 2x^2 + 4x - 8\]
The composition \((g \circ f)(x)\) is defined as \(g(f(x))\). Our objective is to calculate the value of \((g \circ f)(1)\), which is equivalent to finding \(g(f(1))\).
- Evaluating the Inner Function f(1)
To find \((g \circ f)(1)\), we first evaluate the inner function \(f(x)\) at \(x = 1\). Substituting \(x = 1\) into the expression for \(f(x)\):
\[f(1) = -3(1) + 3\]
\[f(1) = -3 + 3\]
\[f(1) = 0\]
So, the inner function evaluates to 0 at \(x = 1\).
- Evaluating the Outer Function g(f(1))
Now that we know \(f(1) = 0\), we substitute this result into the outer function \(g(x)\) to compute \(g(f(1))\), which is \(g(0)\). Substituting \(x = 0\) into the expression for \(g(x)\):
\[g(0) = 2(0)^2 + 4(0) - 8\]
\[g(0) = 2(0) + 0 - 8\]
\[g(0) = 0 + 0 - 8\]
\[g(0) = -8\]
Thus, the value of the composite function \((g \circ f)(1)\) is \(-8\).
Examples
Composite functions are used frequently in scenarios where one process depends on the output of another. For example, imagine you are shopping for a discounted item. Suppose the store offers a 20% discount on an item, and then gives an additional \$5 off coupon. If @@MATH_TOKEN_35@@ represents the price after the 20% discount (@@MATH_TOKEN_36@@) and @@MATH_TOKEN_37@@ represents the price after applying the \$5 coupon (\(x - 5\)), the final price of the item is the composition \(g(f(x))\). By calculating this, you can determine exactly how much you will pay after both processes are applied sequentially!