Question

Find \(f(g(x))\) when \(f(x)=x-7\) and \(g(x)=x^{2}\).

Short answer

Answer: The composition of the functions, \(f(g(x))=x^{2}-7\).

Step-by-step solution

Write the given functions

The functions we are given are \(f(x)=x-7\) and \(g(x)=x^{2}\).

Evaluate \(g(x)\)

Since we are finding \(f(g(x))\), first we need to evaluate \(g(x)\). We have \(g(x)=x^{2}\).

Substitute \(g(x)\) in \(f(x)\)

Now we need to substitute the result of \(g(x)\) in the function \(f(x)\). So, we have \(f(g(x)) = f(x^{2})\).

Evaluate \(f(g(x))\)

Substitute \(x^{2}\) into \(f(x)\): \(f(g(x)) = f(x^{2}) = x^{2}-7\). So, the composition of the functions \(f(x)\) and \(g(x)\), \(f(g(x))=x^{2}-7\).

Key concepts

Composite Functions

In mathematics, composite functions refer to a type of function that is formed when two functions are combined to produce a new function. This new function represents the idea that the output of one function becomes the input of another.

The exercise given involves the composition of two functions, denoted as \(f(g(x))\). To create a composite function, you start with an inner function, such as \(g(x)\), and apply it to a variable \(x\). Then, you take the result of \(g(x)\) and insert it into an outer function, \(f(x)\).

For example, in this exercise, \(f(g(x))\) means that the function \(g\) is applied first and afterward replaced into \(f(x)\).

• Inner function: \(g(x) = x^2\)
• Outer function: \(f(x) = x - 7\).

To find \(f(g(x))\), we calculate \(g(x)\) which gives \(x^2\), and then substitute that expression into \(f(x)\), resulting in \(f(x^2) = x^2 - 7\). This process demonstrates how composite functions operate by nesting one function within another.

Algebraic Functions

Algebraic functions represent a broad class of functions consisting of polynomials, rational functions, and other similar forms. They are defined through algebraic operations such as addition, subtraction, multiplication, division, and exponentiation.

In this problem, both provided functions \(f(x)\) and \(g(x)\) belong to the realm of algebraic functions, specifically polynomials.

• \(g(x) = x^2\) is a quadratic polynomial.
• \(f(x) = x - 7\) is a linear polynomial.

These functions are described using simple expressions, where each function expression follows the conventional rules of algebra.

In the composition, the polynomial properties remain. When \(g(x)\) is used in the composite function \(f(g(x))\), it illustrates the seamless combination of polynomial operations. Algebraic functions provide a straightforward way to model and solve practical problems because of their predictable nature and well-defined operations.

Function Evaluation

Function evaluation is the process of finding the output of a function for specific inputs. It involves substituting a given input into the function and computing the result via the function's formula.

In this exercise, evaluating the composite function \(f(g(x))\) involves a two-step function evaluation process.

First: Evaluate \(g(x)\)

• For an input \(x\), substitute it into \(g(x) = x^2\). This gives a new expression, \(x^2\), as the intermediate result.

Second: Use the result from \(g(x)\) and substitute it into \(f(x)\)

• Take \(x^2\) and substitute it into \(f(x) = x - 7\), resulting in \(f(x^2) = x^2 - 7\).

Function evaluation demonstrates the functional relationship between input values and their respective outputs. It's a crucial skill in mathematics that allows students to solve expressions, model situations, and understand dynamic relationships between variables.