Problem 39 - Algebra 2

Question

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (f \circ g)(c) $$

Step-by-Step Solution

Answer

The expression for \( (f \circ g)(c) \) is \( (c - 3)^2 \).

Step 1: Understanding the given functions and the composite function

The given functions are \( f(x) = x^2 \) and \( g(x) = x - 3 \). The expression we need to compute is \( (f \circ g)(c) \), which translates into \( f(g(c)) \), meaning we substitute \( g(c) \) into the \( f \) function.

Step 2: Substituting \( g(c) \) into \( f \)

Substitute \( g(c) \) into the \( f(x) \) function where \( x \) is located. So, the function \( f(g(c)) \) becomes \( (g(c))^2 \). Since \( g(c) = c - 3 \), we substitute it back into our equation to get \( (c - 3)^2 \).

Step 3: Final Expression

So, our final expression for \( (f \circ g)(c) \) is \( (c - 3)^2 \). Whenever a value for \( c \) is given, it can be substituted in the expression to get a result.

Key Concepts

Quadratic Function

A quadratic function is a type of function that can be written in the form \( f(x) = ax^2 + bx + c \). It features a variable raised to the power of two, making it a polynomial of degree two.
Quadratic functions are significant in mathematics as they describe a parabolic curve that opens either upwards or downwards on a coordinate plane.

• In the given exercise, the specific quadratic function is \( f(x) = x^2 \), which is a simple quadratic function with \( a = 1 \), and \( b = 0 \), \( c = 0 \).
• The graph of this particular function is a parabola that opens upward with its vertex at the origin \((0,0)\).

Quadratic functions can model a wide range of real-world situations, such as the path of a projectile. In our exercise, composing the quadratic function with another function manipulates the input before producing the parabolic output.

Substitution

Substitution is a mathematical technique used to replace a variable in an equation or expression with another expression or value.
It is a vital tool in calculus and algebra, simplifying complex expressions by substituting a manageable component in place of variables.
In the exercise, substitution plays a crucial role in function composition:

• Here, the function \( g(x) = x - 3 \) is substituted into \( f(x) \).
• This means wherever \( x \) appears in \( f(x) = x^2 \), \( g(c) = c-3 \) is substituted in its place.
• This results in \( (c-3)^2 \), modifying the input before applying the quadratic function.

Substitution not only makes expressions manageable but also allows for function compositions, where one function is applied to the result of another.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, operators, and sometimes functions that represent a particular value or set of values.
In the context of our exercise, expressions represent the relationships and transformations between different functions.

• The original expressions \( f(x) = x^2 \) and \( g(x) = x - 3 \) are combined to form a new expression through composition.
• Expression \( (f \circ g)(c) \) turns into \( f(g(c)) \), resulting in a new mathematical expression \( (c-3)^2 \).

Understanding expressions helps in visualizing and solving mathematical concepts. By manipulating expressions, you can evaluate functions at specific points or simplify expressions for easier calculation.