Problem 6 - Calculus with Concepts in Calculus

Question

Let \(f(x)=2 x^{2}+x-4\) and \(g(x)=3-x^{2} .\) Find the specified values. $$ g(f(0)) $$

Step-by-Step Solution

Answer

g(f(0)) = -13

Step 1: Evaluate f(0)

To find \( f(0) \), substitute \( x = 0 \) into the function \( f(x) = 2x^2 + x - 4 \). This gives us: \[ f(0) = 2(0)^2 + 0 - 4 = -4 \] Thus, \( f(0) = -4 \).

Step 2: Evaluate g(f(0))

Now that we know \( f(0) = -4 \), we need to find \( g(f(0)) = g(-4) \). Substitute \( x = -4 \) into the function \( g(x) = 3 - x^2 \). This gives us: \[ g(-4) = 3 - (-4)^2 = 3 - 16 = -13 \] Thus, \( g(f(0)) = -13 \).

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function, distinguished by their highest degree of 2. When graphed, they form a distinctive parabolic shape. A quadratic function generally comes in the form of \( f(x) = ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \).
Quadratics can have various forms:

• Standard form: \( ax^2 + bx + c \)
• Vertex form: \( a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.
• Factored form: \( a(x-p)(x-q) \), where \( p \) and \( q \) are the roots or zeros of the quadratic.

Quadratic functions model many natural phenomena and solve real-world problems.
They often represent scenarios involving projectile motion, optimization problems, and area calculations.

Function Evaluation

Function evaluation entails substituting a given input into a function to find its corresponding output. It's like plugging numbers into a machine and getting an answer.
If you have a function \( f(x) = 2x^2 + x - 4 \) and want to evaluate \( f(0) \), you simply replace every \( x \) with 0.
The calculation would look like:

• First, substitute: \( f(0) = 2(0)^2 + 0 - 4 \)
• Next, simplify: \( f(0) = 0 + 0 - 4 \)
• Result: \( f(0) = -4 \)

This process is repeated whenever you need to evaluate a function at different points, providing valuable insight into how the function behaves.
It's essential when working with any mathematical function because it tells you the output (or value) of a function at a particular input.

Nested Functions

Nested functions involve placing one function inside another. This is often referred to as function composition.
In mathematics, it appears as \( g(f(x)) \), which indicates that you're using the output from \( f(x) \) as the input for \( g(x) \).
Let's break it down with an example:

• Suppose \( f(x) = 2x^2 + x - 4 \) and \( g(x) = 3 - x^2 \).
• First, find \( f(0) \): substitute 0 in \( f(x) \) to get \( -4 \).
• Next, find \( g(f(0)) \) which becomes \( g(-4) \).
• By substituting \( -4 \) into \( g(x) \): \( g(-4) = 3 - (-4)^2 \).
• Simplify: \( g(-4) = 3 - 16 = -13 \).

Function composition, such as nested functions, points to an important concept in mathematics where you 'stack' functions to create complex operations.
This method is useful for breaking down functions into manageable parts and solving layered problems.