Answered

Express your answer as a polynomial in standard form.



$\begin{array}{l}

f(x)=-x-5 \\

g(x)=x^2-6 x+13

\end{array}$



Find: $(f \circ g)(x)$

Verified Answer :

  • Understand that the composition \((f \circ g)(x)\) is equivalent to \(f(g(x))\).
  • Substitute the expression for \(g(x)\) into \(f(x)\): \(-(x^2 - 6x + 13) - 5\).
  • Simplify by distributing the negative sign and combining like terms.
  • The final expression is \(\boxed{-x^2 + 6x - 18}\).

Explanation

  1. Problem Analysis and Objective

We are given two functions, \(f(x) = -x - 5\) and \(g(x) = x^2 - 6x + 13\). Our objective is to find the composition of these two functions, \((f \circ g)(x)\). The composition of functions, denoted by \((f \circ g)(x)\), is defined as evaluating the outer function \(f\) at the output of the inner function \(g(x)\), which is written as \(f(g(x))\). We need to present our final result as a polynomial in standard form, which means writing the terms in descending order of their exponents: \(ax^2 + bx + c\).

  1. Substituting and Simplifying the Composition

To find \((f \circ g)(x)\), we replace the variable \(x\) in the function \(f(x)\) with the entire expression for \(g(x)\).
Given \(f(x) = -x - 5\) and \(g(x) = x^2 - 6x + 13\), we substitute \(g(x)\) into \(f(x)\):
\[(f \circ g)(x) = f(g(x)) = -(x^2 - 6x + 13) - 5.\]
Now, we distribute the negative sign across the terms inside the parentheses:
\[-(x^2 - 6x + 13) = -x^2 + 6x - 13.\]
Finally, we incorporate the constant \(-5\) from the original function \(f(x)\):
\[-x^2 + 6x - 13 - 5.\]

  1. Final Expression and Conclusion

We now combine the constant terms \(-13\) and \(-5\) to simplify the expression further:
\[-13 - 5 = -18.\]
Substituting this back into our expression, we get:
\[(f \circ g)(x) = -x^2 + 6x - 18.\]
The resulting expression is already in standard polynomial form, as the powers of \(x\) are in descending order (\(x^2, x^1, x^0\)). Thus, the final composition is \(\boxed{-x^2 + 6x - 18}\).

Examples

Function composition is widely used in computer programming, where the output of one function serves as the input for another, allowing for modular and efficient code. For example, if you have a function that converts currency (like USD to EUR) and another function that calculates tax on a given amount, you can compose them to calculate the final tax in EUR directly from a USD amount. In this case, applying the tax function to the result of the currency conversion is an example of function composition, simplifying complex multistep processes into a single mathematical operation.

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