Problem 66 - Contemporary Precalculus

Question

The rule of a function \(f\) is given. Write an algebraic formula for \(f(x)\). Triple the input, subtract \(8,\) and take the square root of the result.

Step-by-Step Solution

Answer

Question: Write the algebraic expression representing the function f(x) based on the rule: triple the input, subtract 8, and then take the square root of the result. Answer: f(x) = √(3x - 8)

Step 1: Triple the input

First, we need to triple the input: \(3x\). Here, \(x\) represents the input value.

Step 2: Subtract 8

Next, we need to subtract 8 from the result obtained in Step 1: \(3x - 8\).

Step 3: Take the square root of the result

Lastly, we need to take the square root of the result obtained in Step 2: \(\sqrt{3x - 8}\).

Step 4: Write the algebraic formula for f(x)

Now that we have followed the given rule, we can write an algebraic formula for the function f(x): \(f(x) = \sqrt{3x - 8}\).

Key Concepts

Algebraic Formula

Understanding algebraic formulas is crucial in the realm of mathematics; these formulas are like recipes for solving problems. They are written as an expression or equation representing the relationship between different mathematical symbols and show how to calculate a specific value. In the exercise provided, the goal is to transform a verbal rule into an algebraic formula. When we are given a sequence of operations to apply to an input, such as 'triple the input, subtract 8, and take the square root', we perform these actions step by step.

First, the input, signified by the variable 'x', is tripled, resulting in the term '3x'. After that, 8 is subtracted from this result, altering the expression to '3x - 8'. The final instruction is to take the square root of the outcome, which is expressed by placing the previous expression under a square root sign, so we get \(\sqrt{3x - 8}\). This concise expression is the algebraic formula that describes the function's rule and illustrates the power of algebra in condensing a series of actions into a form that is easy to work with mathematically.

Function Notation

Function notation is the way in which a function is denoted and it's how we communicate the idea of a function mathematically. Rather than saying 'the output when you input x into the function', we use the shorthand f(x) to express the same concept. Simplifying communication and making the relationship between input (x) and output (f(x)) clear is essential.

Using function notation serves several important purposes: It names the function, in this case, 'f', allowing us to differentiate between multiple functions in a problem. It also shows what variable is being used, which tells us what the input is. In the example \(f(x) = \sqrt{3x - 8}\), 'f' is the function name, 'x' is the input, and \(\sqrt{3x - 8}\) is the formula or rule that the function follows to transform 'x' into the output. This notation is compact, efficient, and widely used in algebra and calculus for representing functions and assisting in function evaluation and analysis.

Square Root Operations

Square root operations are a fundamental concept in algebra that deal with finding a number which, when multiplied by itself, gives the original number under the square root. The square root of a number 'a' is denoted as \(\sqrt{a}\). Understanding how to handle square roots can be pivotal in simplifying expressions and solving equations.

When dealing with algebraic expressions like the one in our exercise, \(\sqrt{3x - 8}\), there are several key points to remember: The value under the square root, here '3x - 8', is called the radicand, and it cannot be negative if we are dealing with real numbers because the square root of a negative number is not defined in the real number system. Additionally, when performing operations involving square roots, we must always respect the order of operations, which in the case of our exercise, means we triple 'x' and subtract '8' before taking the square root. It's also important to note that squaring a square root, such as \(\left(\sqrt{a}\right)^2\), will simply give us the original number 'a' under the square root—a useful fact when solving equations.