Problem 14 - 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. $$(f g)(-3)$$

Step-by-Step Solution

Answer

(fg)(-3) = 28

Step 1: Understand Composition of Functions

The composition of functions \((fg)(x)\) means we first apply the \(g(x)\) function and then substitute it into the \(f(x)\) function. This is expressed as \(f(g(x))\).

Step 2: Compute g(-3)

Substitute \(x = -3\) into the function \(g(x) = 2x - 1\):\[g(-3) = 2(-3) - 1 = -6 - 1 = -7\]So, \(g(-3) = -7\).

Step 3: Substitute g(-3) into f(x)

Substitute \(g(-3) = -7\) into \(f(x) = x^2 + 3x\):\[f(-7) = (-7)^2 + 3(-7)\]Simplify it to calculate \(f(-7)\):\[f(-7) = 49 - 21 = 28\]

Step 4: Write the Final Result

Therefore, the result of \((fg)(-3)\) is \(f(-7) = 28\).

Key Concepts

Quadratic Function

Quadratic functions are a type of polynomial function where the highest degree of the variable is 2. In simpler terms, these functions can be written in the standard form: \[ f(x) = ax^2 + bx + c \]where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The graph of a quadratic function is a parabola. Parabolas have unique properties:

• A "vertex" which is the highest or lowest point, depending on whether the parabola opens upwards or downwards.
• An "axis of symmetry" that is a vertical line through the vertex.
• They can open up or down depending on the sign of \(a\). Positive \(a\) means it opens upwards and negative \(a\) means it opens downwards.

In our exercise, the function \(f(x) = x^2 + 3x\) is a quadratic function. Here, \(a = 1\), \(b = 3\), and \(c = 0\). This function graphs as a parabola opening upwards because \(a\) is positive. Understanding how these functions behave will help you predict graph shapes and solve real-world problems. When working with quadratic functions, anticipate tasks like finding vertices, intercepts, and determining the direction of the graph. By mastering these attributes, you're equipped to handle various applications from physics to finance.

Linear Function

Linear functions are the foundations of algebra, simple yet fundamental in mathematics. A linear function is a polynomial of the first degree and is expressed in the form:\[ g(x) = mx + b \]where \(m\) is the slope and \(b\) is the y-intercept. The graph of a linear function is a straight line:

• The "slope" \(m\) indicates the steepness of the line and its direction (positive slope for rising lines and negative for falling lines).
• The "intercept" \(b\) is where the line crosses the y-axis.

In our given problem, the function \(g(x) = 2x - 1\) is a perfect example of a linear function. Here, \(m = 2\) indicates that for every unit increase in \(x\), \(g(x)\) increases by 2 units. Meanwhile, \(b = -1\) implies that the line crosses the y-axis at \(-1\).Understanding linear functions is crucial as they often appear in various situations, from calculating speeds to predicting profits in business. They represent constant rates of change and pose fewer complications compared to higher-degree polynomials. Mastering these functions paves the way for exploring more complex mathematical models.

Polynomials

Polynomials are an essential class of functions that form the backbone of algebra. They are expressions built from variables and constants using only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.An example of a polynomial is any function in the form:\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \]Here, \(a_n, a_{n-1}, \ldots, a_0\) are coefficients, and \(n\) is a non-negative integer representing the highest degree, which dictates the polynomial's general behavior and shape of its graph. Different polynomial expressions include:

• Linear Polynomials: Of degree 1, like \(g(x) = 2x - 1\).
• Quadratic Polynomials: Of degree 2, such as \(f(x) = x^2 + 3x\).
• Constant Polynomials: Degree 0, which are values like \(P(x) = 5\).

The beauty of polynomials lies in their versatility and simplicity. They can model a wide range of situations by adjusting their coefficients and degrees. Understanding the basics of polynomials aids in grasping more complex concepts such as function transformations, integrations, and in solving equations, which are prevalent across all branches of mathematics and applicable in technology, economics, and the sciences. By knowing how to decompose complex situations into polynomial functions, you open doors to understanding a large part of mathematical analysis.