Answered

Express your answer as a polynomial in standard form.



$\begin{array}{l}

f(x)=x+2 \\

g(x)=3 x^2+6 x-8

\end{array}$



Find: $g(f(x))$

Verified Answer :

  • Identify the composition \(g(f(x)) = g(x+2)\) by substituting \(f(x)\) into \(g(x)\).
  • Expand the resulting expression \(3(x+2)^2 + 6(x+2) - 8\).
  • Distribute the coefficients to get \(3x^2 + 12x + 12 + 6x + 12 - 8\).
  • Combine like terms to obtain the final polynomial: \(\boxed{3x^2 + 18x + 16}\).

Explanation

  1. Analysis of the Problem

We are given two functions: \(f(x) = x + 2\) and \(g(x) = 3x^2 + 6x - 8\). Our objective is to determine the composition \(g(f(x))\), which involves substituting the entire expression for \(f(x)\) into the variable \(x\) within the function \(g(x)\). We then simplify the resulting expression to obtain a polynomial in standard form \(ax^2 + bx + c\).

  1. Substituting and Expanding the Terms

To find \(g(f(x))\), we substitute \(x + 2\) for every instance of \(x\) in the definition of \(g(x)\):
\[g(f(x)) = g(x + 2) = 3(x + 2)^2 + 6(x + 2) - 8.\]
Next, we expand the squared binomial \((x + 2)^2\), which results in \(x^2 + 4x + 4\). Substituting this back into our equation gives:
\[g(f(x)) = 3(x^2 + 4x + 4) + 6(x + 2) - 8.\]

  1. Distributing Coefficients

Now, we distribute the coefficients into the parentheses:

  • For the first term: \(3 \times (x^2 + 4x + 4) = 3x^2 + 12x + 12\).
  • For the second term: \(6 \times (x + 2) = 6x + 12\).

Substituting these into the expression:
\[g(f(x)) = 3x^2 + 12x + 12 + 6x + 12 - 8.\]

  1. Simplifying and Final Result

Finally, we group and combine the like terms:

  • The \(x^2\) term remains \(3x^2\).
  • The \(x\) terms are \(12x + 6x = 18x\).
  • The constant terms are \(12 + 12 - 8 = 16\).

Combining these results, we get the final polynomial in standard form:
\[g(f(x)) = 3x^2 + 18x + 16.\]
Thus, the final answer is \(\boxed{3x^2 + 18x + 16}\).

Examples

Function composition is widely used in computer science and programming, where one function's output serves as the input to another. For instance, if you have a program that first converts a measurement from inches to centimeters (a linear transformation \(f(x) = 2.54x\)) and a second program that calculates the surface area of a circle based on a given radius in centimeters (a quadratic transformation \(g(r) = \pi r^2\)), calculating the composition \(g(f(x))\) allows you to determine the area directly from the inch input. This modular approach simplifies complex calculations by breaking them down into manageable, reusable steps.

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