Problem 52 - Brief Applied Calculus

Question

For each pair of functions \(f(x)\) and \(g(x)\), find a. \(f(g(x))\) b. \(g(f(x))\) and c. \(f(f(x))\) $$ f(x)=\sqrt{x} ; \quad g(x)=x^{3}-1 $$

Step-by-Step Solution

Answer

a. \(f(g(x)) = \sqrt{x^3 - 1}\) b. \(g(f(x)) = x^{3/2} - 1\) c. \(f(f(x)) = x^{1/4}\)

Step 1: Define f(g(x))

To find \(f(g(x))\), substitute \(g(x)\) into \(f\). Since \(f(x) = \sqrt{x}\) and \(g(x) = x^3 - 1\), we substitute \(x^3 - 1\) into \(f\), giving us \(f(g(x)) = \sqrt{x^3 - 1}\).

Step 2: Define g(f(x))

Next, we need to find \(g(f(x))\) by substituting \(f(x)\) into \(g\). Given \(g(x) = x^3 - 1\), substitute \(f(x) = \sqrt{x}\) into \(g\): \(g(f(x)) = (\sqrt{x})^3 - 1 = x^{3/2} - 1\).

Step 3: Define f(f(x))

Finally, we find \(f(f(x))\) by substituting \(f(x)\) into itself. Since \(f(x) = \sqrt{x}\), substitute it to get \(f(f(x)) = f(\sqrt{x}) = \sqrt{\sqrt{x}} = x^{1/4}\).

Key Concepts

Domain and Range

The domain and range of a function refer to the set of all possible input values (domain) and the set of all possible output values (range). Understanding these concepts is crucial, especially when working with square root functions and composite functions.

• **Domain**: For the square root function \( f(x) = \sqrt{x} \), the domain comprises non-negative numbers. So, \( x \ge 0 \). In contrast, for the function \( g(x) = x^3 - 1 \), the domain is all real numbers since cubic functions accept all real inputs.


• **Range**: For \( f(x) = \sqrt{x} \), the range includes only non-negative numbers. This means \( f(x) \ge 0 \). For \( g(x) \), given any real number input, we can get any real number as output. Thus, the range of \( g(x) \) is also all real numbers.


• **Composite Functions**: When dealing with compositions like \( f(g(x)) \) or \( g(f(x)) \), reassessing the domain and range is vital. For instance, for \( f(g(x)) = \sqrt{x^3 - 1} \), the domain must ensure that the expression under the square root is non-negative, i.e., \( x^3 - 1 \ge 0 \). This affects what input values are accepted.

Composition of Functions

The composition of functions involves applying one function to the results of another. It creates a new function whose output depends on the nested functions' interaction. Understanding composition helps solve problems where individual function outputs become the next function's input.

• **Notation**: Typically, function composition is denoted by \( (f \circ g)(x) = f(g(x)) \). It implies applying \( g(x) \) first, followed by \( f(x) \).


• **Process**: To compute compositions like \( f(g(x)) \) or \( g(f(x)) \), you substitute the output of one function directly into the next function. For \( f(g(x)) = \sqrt{x^3 - 1} \), it means placing \( g(x) = x^3 - 1 \) inside \( f(x) = \sqrt{x} \), effectively nesting one function inside the other.


• **Order Matters**: It’s essential to remember the order of operations in composition changes the outcome. \( f(g(x)) \) yields a different result than \( g(f(x)) \). As shown above, \( f(g(x)) \) involves a square root of a cubic function, while \( g(f(x)) \) cubes the square root of \( x \), illustrating how outputs vary depending on the nesting order.

Square Root Functions

Square root functions, like \( f(x) = \sqrt{x} \), are pivotal in various mathematical contexts due to their unique properties.

• **Definition**: A square root function outputs the non-negative root of its input. These functions grow slower since they increase with the square root of \( x \).


• **Graph**: Typically, the graph of \( y = \sqrt{x} \) forms a curve that extends rightward and upward from the origin. It reflects that the function only accepts non-negative values for \( x \).


• **Behaviour in Composition**: When combined in compositions like \( f(f(x)) = \sqrt{\sqrt{x}} = x^{1/4} \), square root functions can exhibit even slower growth, emphasizing the implications of nesting roots. Simplifying such nested roots requires understanding power rules since double roots imply taking the power of \( x \) to a fraction of a fraction, i.e., \( x^{1/4} \).