Problem 9 - A Graphical Approach to Precalculus with Limits

Question

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$(f \circ g)(x)$$

Step-by-Step Solution

Answer

The composition \((f \circ g)(x) = 4x^2 + 2x - 2\).

Step 1: Understand Function Composition

Function composition \((f \circ g)(x)\) means you need to apply one function to the results of another. Here, \(g(x)\) is applied first, and then \(f(x)\) is applied to that result. So, it's equivalent to \(f(g(x))\).

Step 2: Substitute g(x) into f(x)

To find \(f(g(x))\), replace every \(x\) in \(f(x) = x^2 + 3x\) with \(g(x) = 2x - 1\). So, \(f(g(x)) = (2x - 1)^2 + 3(2x - 1)\).

Step 3: Simplify the Quadratic Part

Calculate \((2x - 1)^2\) which expands to \(4x^2 - 4x + 1\) using the identity \((a - b)^2 = a^2 - 2ab + b^2\).

Step 4: Simplify the Linear Part

Calculate \(3(2x-1) = 6x - 3\).

Step 5: Combine and Simplify

Combine \((2x - 1)^2\) and \(3(2x - 1)\) to get the full expression: \((4x^2 - 4x + 1) + (6x - 3) = 4x^2 + 2x - 2\).

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function where the highest degree is two. The general form of a quadratic function is expressed as \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Quadratics graph as a parabola, which can open upwards or downwards depending on the sign of \( a \).

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), it opens downwards.

The peak or the lowest point of the parabola is known as the vertex, which provides vital information about the graph's direction and location.
The standard method to find this point is by completing the square or using the vertex formula, \( x = -\frac{b}{2a} \).
In solving problems involving quadratic functions within compositions like \( f \circ g \), essential skills include expanding squares, recognizing patterns, and simplifying expressions.

Polynomial Functions

Polynomial functions are one of the most fundamental types of algebraic expressions, which can take different forms based on the highest power of the variable present. They are expressed as \( a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \), where \( n \) is a non-negative integer, and \( a_n eq 0 \).

• Linear Polynomial: Power of one, \( ax + b \)
• Quadratic Polynomial: Power of two, \( ax^2 + bx + c \)
• Cubic Polynomial: Power of three, \( ax^3 + bx^2 + cx + d \)

The degree of a polynomial is essential as it indicates the number of potential roots and the basic shape of its graph.
Polynomial functions can be added, subtracted, multiplied, and composed. When performing composition, like in \( f(g(x)) \), it involves substituting one polynomial into another and then simplifying the results by combining like terms and arranging in standard form.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They form the building blocks of polynomial and rational expressions.
When working with algebraic expressions in the context of function operations such as \( (f \circ g)(x) \), simplification is a frequent task that involves manipulating terms to make expressions more understandable or to find specific information.

• Combine like terms to reduce complexity.
• Use distributive property to expand expressions.
• Factor expressions when needed to simplify.

These techniques are vital when undertaking function compositions, as they help in breaking down complex problems into more manageable steps, like transforming \( (2x-1)^2 + 3(2x-1) \) into a simplified form \( 4x^2 + 2x - 2 \). Consistent practice in manipulating algebraic expressions will enhance problem-solving skills and mathematical understanding.