Problem 24 - Precalculus with Limits

Question

In Exercises 17-28, evaluate the indicated function for \(f(x) = x^2 + 1\) and \(g(x) = x - 4\). \((fg)(-6)\)

Step-by-Step Solution

Answer

The value of \((fg)(-6)\) is 101.

Step 1: Evaluate g(x) at x = -6

We start by finding the value of \(g(x)\) at \(x = -6\). In simpler terms, we need to calculate \(g(-6)\). Considering \(g(x) = x - 4\), we replace \(x\) with \(-6\), giving us \(-6 - 4 = -10\).

Step 2: Evaluate the composite function (fg)(-6)

Now, we evaluate the composite function by substituting \(g(-6)\) into \(f(x)\). Which means to calculate \(f(g(-6))\). Since we know from the previous step that \(g(-6) = -10\), we substitute \(-10\) into \(f(x)\). With \(f(x)\) given as \(x^2 + 1\), we therefore have \((-10)^2 + 1 = 101\).

Key Concepts

Composite Functions

Composite functions involve combining two functions together to form a new function. Here's how it works: you first apply one function to an input, then take the result and apply another function to it. It's like a function of a function. In math notation, a composite function using functions \( f \) and \( g \) is expressed as \((f \circ g)(x) = f(g(x))\). This means you first calculate \( g(x) \), and then use the result as the input for \( f(x) \).
Let's look at how this applies to the exercise:

• You start with the second function \( g(x) = x - 4 \). First, plug in your value into this function. For \( x = -6 \), you get \( g(-6) = -10 \).
• Take \( -10 \) and plug it into the first function \( f(x) = x^2 + 1 \). This gives \( f(-10) = (-10)^2 + 1 = 101 \).

Each step involves substitution and solving, which forms the basis of evaluating composite functions.

Quadratic Functions

Quadratic functions are a type of polynomial function where the highest exponent of the variable is 2. They have the form \( f(x) = ax^2 + bx + c \). In this exercise, the function \( f(x) = x^2 + 1 \) is a simple quadratic function.
Quadratics have several key features:

• The graph of a quadratic function is a parabola. For \( f(x) = x^2 + 1 \), the parabola opens upwards.
• The vertex is the point on the parabola where it changes direction. In the function \( x^2 + 1 \), the vertex is at (0,1) since it's the minimum point on this graph.
• They also have axes of symmetry. The axis is a vertical line passing through the vertex.

Quadratic functions are commonly dealt with in many mathematical situations, and their evaluation often involves squaring numbers and adding constants.

Algebraic Operations

Algebraic operations are the core processes in algebra and consist of addition, subtraction, multiplication, and division. These operations are used to combine numbers or variables in equations.
In the step-by-step solution provided for the problem, you can observe some key algebraic operations:

• Finding \( g(-6) \) involves subtraction: \( -6 - 4 \) equals \(-10\). Here, subtraction removes a value from another.
• Evaluating \( f(-10) \) uses both squaring, which is a form of multiplication \((-10 \times -10)\), and addition: \(100 + 1\) resulting in \(101\). These operations form the basis of solving the quadratic function \((x^2 + 1)\).

Understanding these basic operations greatly aids in solving more complex problems in mathematics, like function evaluation. Mastering them is foundational for progress in algebra and beyond.