Problem 18 - Calculus for Biology and Medicine

Question

Suppose that $$ f(x)=x^{4}, \quad x \geq 3 $$ and $$ g(x)=\sqrt{x+1}, \quad x \geq 3 $$ Find \((f \circ g)(x)\) together with its domain.

Step-by-Step Solution

Answer

The function \((f \circ g)(x) = (x+1)^2\) with domain \(x \geq 3\).

Step 1: Understand Function Composition

The composition of two functions \((f \circ g)(x)\) means that we will substitute \(g(x)\) into \(f(x)\). This operation is read as "\(f\) of \(g\) of \(x\)".

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

Given that \(g(x) = \sqrt{x+1}\), substitute \(g(x)\) into \(f(x) = x^4\). This gives us:\[(f \circ g)(x) = f(g(x)) = f(\sqrt{x+1}) = (\sqrt{x+1})^4\] which simplifies to \((x+1)^2\).

Step 3: Check the Domain of g(x)

The domain of \(g(x)\) is given as \(x \geq 3\) because for the function \(\sqrt{x+1}\) to be defined, \(x+1\) must be positive. Subtracting 1 from both sides of \(x+1 \geq 0\) results in \(x \geq -1\), but due to the initial condition of \(x \geq 3\), the effective domain is \(x \geq 3\).

Step 4: Confirm the Domain of f(g(x))

For the composition \((f \circ g)(x) = (x+1)^2\), since \(x+1\) is a polynomial which is always defined for any real number, the domain of \((f \circ g)\) is restricted by \(g(x)\)'s domain. Hence, the domain of \((f \circ g)(x)\) is also \(x \geq 3\).

Step 5: Conclude the Result

The function \((f \circ g)(x)\) is \((x+1)^2\) with a domain of \(x \geq 3\).

Key Concepts

Domain of a Function

When you're working with functions, understanding the domain is very important. The domain is the set of all possible input values (usually represented by "x") that a function can accept without causing any mathematical problems. For example, in the function \( f(x) = \sqrt{x+1} \), the expression under the square root, \( x+1 \), must be non-negative. This is because you can't find the square root of a negative number within the real numbers. Hence, for \( g(x) = \sqrt{x+1} \), we solve the inequality:

• \( x+1 \geq 0 \)
• \( x \geq -1 \)

However, the problem states that \( x \geq 3 \), which further restricts the domain. Therefore, the domain of \( g(x) \) starts at 3. Similarly, when finding the domain of \( (f \circ g)(x) \), it is mainly restricted by \( g(x) \)'s domain, as \( f(x) = x^4 \) can accept any real number. So, the domain of \( (f \circ g)(x) \) is \( x \geq 3 \) too.

Polynomial Function

A polynomial function is one of the most straightforward types of function you can encounter. It involves expressions containing variables raised to whole number powers, possibly with coefficients. Consider \( f(x) = x^4 \) as an example, which is a polynomial of degree 4. This type of function has some nice properties:

• It's continuous, meaning there are no breaks, jumps, or holes in its graph.
• It's defined for all real numbers, unlike some other types of functions.

When you perform function composition, such as finding \( (f \circ g)(x) \), the outcome may look like another polynomial. In this case, substituting \( g(x) = \sqrt{x+1} \) into \( f(x) \) turns into \( (x+1)^2 \), a quadratic polynomial function. This new polynomial also inherits the continuity and real number domain characteristics inherent in polynomials.

Radical Function

Radical functions typically involve a square root \((\sqrt{\;})\), cube root, or higher roots. These can lead to restrictions on the domain since roots require certain conditions to hold within the real numbers. Let's explore the function \( g(x) = \sqrt{x+1} \). Here, ensuring that \( x+1 \geq 0 \) is crucial, because a negative value under a square root would not return a real number.
In the original exercise, we are told \( x \geq 3 \), ensuring the value under the radical is always positive. Radical functions can sometimes make compositions trickier, as they impose their domain restrictions on the outcome. During function composition with polynomials like in \( (f \circ g)(x) \), these domain constraints remain essential. Hence, understanding the domain of radical components is important whenever you're dealing with compositions involving radical expressions.