Problem 22 - Calculus Single Variable

Question

Let \(f(x)=\sqrt{2 x+5},\) and \(g(x)=x^{-1 / 3} .\) In Exercises \(19-22\) calculate the given expression. \((g \circ g)(512)\)

Step-by-Step Solution

Answer

The result is 2.

Step 1: Understand the Composition

The composition \(g \,g\)\(x\) means applying the function \(g\) to the input, and then again apply \(g\) to the result. Symbolically, \(g \, g(x) = g(g(x))\).

Step 2: Apply g to x

The function \(g(x) = x^{-1/3}\). First, we need to find \(g(512)\). Given that \(512 = 8^3\), we have:\[g(512) = 512^{-1/3} = (8^3)^{-1/3} = 8^{-1} = \frac{1}{8} \]

Step 3: Apply g to g(x)

Now apply \(g\) again to the result \(\frac{1}{8}\) obtained from Step 2. This means we need to find \(g\left(\frac{1}{8}\right)\): Given \(\frac{1}{8} = 2^{-3}\), we have:\[g\left(\frac{1}{8}\right) = \left(\frac{1}{8}\right)^{-1/3} = (2^{-3})^{-1/3} = 2^{-3 \times (-1/3)} = 2^{-(-1)} = 2^1 = 2 \]

Key Concepts

Inverse Function

An inverse function essentially undoes the action of another function. If you have a function that takes an input and produces an output, the inverse function takes that output and returns it back to the original input. For example, if a function \( f(x) = y \), then its inverse \( f^{-1}(y) \) will yield \( x \) when you input \( y \). It's like retracing your steps.

To have an inverse, a function must be one-to-one, which means each output is from a distinct input. A classic example of inverse functions is the pair of exponentiation and logarithm functions, where the logarithm "undoes" the effect of exponentiation. Visualizing these functions, you'll notice they reflect over the line \( y = x \) when plotted graphically. This visualizes the concept that each accessible input-output pair is perfectly reversible.

Exponentiation

Exponentiation is the process of raising a number to the power of another number. It is represented by the notation \( a^b \), where \( a \) is the base and \( b \) is the exponent. This operation repeats multiplication of the base \( a \) by itself \( b \) times.

For instance, \( 2^3 = 2 \times 2 \times 2 = 8 \). It serves a crucial role in various mathematical operations, appearing frequently in exponential functions, growth modelling, and the analysis of algorithms.

Exponentiation has interesting properties:

• The power of zero: Any non-zero number raised to the power of zero is equal to one (\( a^0 = 1 \), \( a eq 0 \)).
• The power of one: A number raised to the power of one is the number itself (\( a^1 = a \)).
• Negative exponents: They represent the reciprocal of the base raised to the corresponding positive exponent (\( a^{-b} = \frac{1}{a^b} \)).
• Fractional exponents: \( a^{1/n} \) results in the \( n \)-th root of \( a \), bridging the concept of roots and powers.

Square Root Function

The square root function is a special type of root function, where the power is \( 1/2 \). When you see the notation \( \sqrt{x} \), it's equivalent to \( x^{1/2} \). The square root of a number is a value that, when multiplied by itself, gives the original number.

For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \). Square roots are often used to find dimensions in geometry or to simplify equations in algebra. They help solve quadratic equations, where understanding the square's foundations is essential.

Key points:

• Non-negative roots for real numbers: The principal square root function is defined for non-negative numbers only in the real numbers domain (\( x \geq 0 \)).
• Connection to the quadratic equation: The square root function helps in solving equations of the form \( x^2 = a \), offering solutions as \( x = \pm \sqrt{a} \).
• Graphical representation: The function \( y = \sqrt{x} \) forms half of a parabola on its side on the coordinate plane.

Mathematical Notation

Mathematical notation involves a system of symbols used to represent numbers, functions, operations, and relationships in mathematics. Notation helps convey complex ideas succinctly, offering clarity and precision.

Understanding notation is crucial for solving equations, communicating mathematical thoughts, and conducting proofs effectively.

Common notations include:

• Function notation: \( f(x) \) denotes a function named \( f \), taking \( x \) as input and returning an output.
• Exponentiation: The notations \( a^b \) or \( x^{-1/3} \) represent powers and roots, aiding transformation and simplification of expressions.
• Composite functions: The notation \( (g \circ g)(x) \) indicates applying function \( g \) twice, showcasing composition.
• Root notation: \( \sqrt{x} \) and \( x^{1/2} \) for square roots, extending to \( x^{1/n} \) for other roots.

By mastering these notations, mathematical problems become approachable and easier to navigate. Familiarization with such symbols and their interactions is essential for anyone studying mathematics at any level.