Problem 41 - Algebra and Trigonometry

Question

\(41-44\) Find \(f \circ g \circ h\) $$ f(x)=x-1, \quad g(x)=\sqrt{x}, \quad h(x)=x-1 $$

Step-by-Step Solution

Answer

The composition is \( f \circ g \circ h = \sqrt{x-1} - 1 \).

Step 1: Understand the Composition

We need to find the composition of functions in the order \( f \circ g \circ h \). This means we will first apply \( h \), then \( g \), and finally \( f \) to the function.

Step 2: Find \( h(x) \)

From the function definition, \( h(x) = x - 1 \). We apply this to our input \( x \) to get the expression \( x - 1 \).

Step 3: Find \( g(h(x)) \)

Substitute \( h(x) = x - 1 \) into \( g \). Thus, \( g(h(x)) = g(x-1) = \sqrt{x-1} \).

Step 4: Find \( f(g(h(x))) \)

Substitute \( g(h(x)) = \sqrt{x-1} \) into \( f \). So, \( f(g(h(x))) = f(\sqrt{x-1}) = \sqrt{x-1} - 1 \).

Step 5: Combine Functions

The composition \( f \circ g \circ h \) results in the function \( f(g(h(x))) = \sqrt{x-1} - 1 \).

Key Concepts

Composing Functions

Function composition is like a set of Russian nesting dolls. We take one function and "nest" or "compose" it inside another. When we talk about composing functions such as \( f \circ g \circ h \), we apply them in a specific order. First, we use the innermost function, and then work our way outwards. The goal is to create a new function that combines the effects of each of the individual functions. For example, if we have three functions:

• \( f(x) = x - 1 \)
• \( g(x) = \sqrt{x} \)
• \( h(x) = x - 1 \)

We find the composition \( f \circ g \circ h \) by: - Applying \( h \) to our input first, giving us \( h(x) = x - 1 \) - Feeding the result into \( g \), thus \( g(h(x)) = \sqrt{x - 1} \) - Finally, we plug this into \( f \), yielding \( f(g(h(x))) = \sqrt{x-1} - 1 \). Function composition allows us to combine multiple transformations into a single step, simplifying complex operations.

Order of Operations

When working with function compositions, it's crucial to follow the correct order of operations. Similar to arithmetic operations, the sequence in which we apply functions affects the outcome. Remember, with function composition, we always start from the innermost function and move outward. This is the opposite of arithmetic, where we usually resolve from left to right once inside parentheses.In the example of the composition \( f \circ g \circ h \):

• We start with \( h(x) = x - 1 \)
• Then move to \( g(h(x)) = \sqrt{x - 1} \)
• And finally \( f(g(h(x))) = \sqrt{x-1} - 1 \)

Keeping track of this order ensures clarity and prevents mistakes. Remembering this flow helps in dealing with nested functions and makes complex operations more manageable.

Function Notation

Function notation is a concise way of expressing mathematical operations, helping us to succinctly capture relations between input and output. When we write \( f(x) \), the letter \( f \) represents the function's name, and \( x \) is the input variable. This notation helps clarify which function is being applied, and what expression is involved.In composition such as \( f \circ g \circ h \), this notation becomes especially helpful:

• Each function \( f \), \( g \), and \( h \) gets applied with its domain of input set by the outcome of the previous function.
• By writing each step, like \( f(g(h(x))) \), we can make sure we're keeping the logical order in check.
• It also provides a clear symbolic representation that aids in both communication and further manipulation.

Function notation is a fundamental tool in mathematics, enabling us to build, deconstruct, and analyze functions efficiently.