Problem 4 - Applied Calculus

Question

For the functions \(f\) and \(g\), find (a) \(f(g(1))\) (b) \(g(f(1))\) (c) \(f(g(x))\) (d) \(g(f(x))\) (e) \(f(t) g(t)\) $$f(x)=\sqrt{x+4}, g(x)=x^{2}$$

Step-by-Step Solution

Answer

(a) \(\sqrt{5}\); (b) 5; (c) \(\sqrt{x^2+4}\); (d) \(x+4\); (e) \(\sqrt{t+4} \cdot t^2\).

Step 1: Understand the Functions

First, identify the given functions. We have \( f(x) = \sqrt{x + 4} \) and \( g(x) = x^2 \). Understand that \( f \) involves a square root and \( g \) is a quadratic function.

Step 2: Calculate f(g(1))

Evaluate \( g(1) \) first: \( g(1) = 1^2 = 1 \). Then use the result to find \( f(g(1)) = f(1) = \sqrt{1 + 4} = \sqrt{5} \).

Step 3: Calculate g(f(1))

First, calculate \( f(1) = \sqrt{1 + 4} = \sqrt{5} \). Then, evaluate \( g(f(1)) = g(\sqrt{5}) = (\sqrt{5})^2 = 5 \).

Step 4: Express f(g(x))

Compute \( f(g(x)) \) by substituting \( g(x) \) into \( f \): \( f(g(x)) = f(x^2) = \sqrt{x^2 + 4} \).

Step 5: Express g(f(x))

Compute \( g(f(x)) \) by substituting \( f(x) \) into \( g \): \( g(f(x)) = g(\sqrt{x + 4}) = (\sqrt{x + 4})^2 = x + 4 \).

Step 6: Calculate f(t)g(t)

Find \( f(t)g(t) \) by multiplying the functions: \( f(t)g(t) = \sqrt{t + 4} \cdot t^2 \).

Key Concepts

Quadratic Function

A quadratic function is a fundamental concept in algebra characterized by a variable raised to the power of two, often written in the form of \( g(x) = ax^2 + bx + c \). In the given exercise, the quadratic function is represented as \( g(x) = x^2 \). This is a basic form where the coefficient \( a = 1 \), and \( b \) and \( c \) are zero, making it a simple parabola that opens upwards.
Quadratic functions have several interesting properties:

• The graph of a quadratic function is always a parabola.
• The axis of symmetry can be found using the formula \( x = -\frac{b}{2a} \).
• The vertex, the highest or lowest point, offers valuable insights about maximum or minimum values of the function.
• For \( g(x) = x^2 \), the parabola is symmetric about the y-axis.

Understanding the behavior of quadratic functions is crucial, especially when evaluating them or using them in composite functions.

Square Root Function

A square root function is another vital mathematical function defined by the form \( f(x) = \sqrt{x} \). In this exercise, we're given \( f(x) = \sqrt{x + 4} \). Considering the domain of the square root function is particularly important since you can only take the square root of non-negative numbers.
The properties of the square root function are:

• Its domain depends on the expression inside the square root. Here, \( x + 4 \geq 0 \), or \( x \geq -4 \).
• The range of a simple square root function \( \sqrt{x} \) is all non-negative numbers.
• The graph typically looks like a curve starting from a point on the x-axis and increasing to the right.

Understanding these characteristics helps in evaluating and composing the function with others, ensuring we respect valid input values to avoid mathematical errors.

Evaluating Functions

Evaluating functions involves substituting a given input into the function and performing arithmetic operations to find the output. With our functions \( f(x) = \sqrt{x + 4} \) and \( g(x) = x^2 \), you evaluate them by replacing \( x \) with a specific number or expression.
Here’s how we approached evaluating in the exercise:

• To find \( g(1) \), substitute 1 for \( x \) in \( g(x) \), yielding \( 1^2 = 1 \).
• For \( f(1) \), replace \( x \) with 1 in \( f(x) \), resulting in \( \sqrt{1 + 4} = \sqrt{5} \).

Evaluating is key to understanding specific outputs and is foundational for more complex operations like finding composite functions.

Composite Functions

Composite functions build upon basic function evaluations by combining two functions into one. They answer the question, "What is the result when one function is applied to the outcome of another function?" For our functions \( f(x) = \sqrt{x + 4} \) and \( g(x) = x^2 \), composite functions like \( f(g(x)) \) involve substituting the entire function \( g(x) \) into \( f(x) \).
Understanding composite functions includes:

• To find \( f(g(x)) \), substitute \( g(x) = x^2 \) into \( f(x) \), resulting in \( \sqrt{x^2 + 4} \).
• For \( g(f(x)) \), replace \( x \) in \( g \) with \( f(x) = \sqrt{x + 4} \), leading to \( (\sqrt{x + 4})^2 = x + 4 \).
• Composite functions help explore how different transformations and operations affect expressions when combined.

Mastering composite functions allows for deeper insights into function behaviors and is a crucial skill in advanced mathematics.