Function Formula From Verbal Description Step By Step Guide

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In mathematics, a function is often described verbally, outlining the steps involved in transforming an input value into an output value. Translating these verbal descriptions into algebraic formulas is a fundamental skill in mathematics. This article delves into the process of converting a verbal description of a function into its algebraic representation, providing a step-by-step guide and illustrating the process with a specific example. Let's consider the verbal description of a function ff: "Subtract 4, then cube the result." Our goal is to find a formula that expresses this function algebraically, allowing us to represent f(x)f(x).

Decoding the Verbal Description

Before we can write an algebraic formula, we need to carefully dissect the verbal description and identify the operations involved and their order. The description "Subtract 4, then cube the result" clearly indicates two operations:

  1. Subtraction: We begin by subtracting 4 from the input value.
  2. Cubing: After subtracting 4, we cube the result of the subtraction.

The order of these operations is crucial. We must perform the subtraction before cubing the result. This understanding is paramount in constructing the correct algebraic formula.

Building the Algebraic Formula

Now that we understand the operations and their order, we can translate them into an algebraic formula. Let's break down the process:

  1. Represent the input: We use the variable xx to represent the input value to the function ff.

  2. Subtract 4: The first operation is to subtract 4 from the input xx. This can be written as x−4x - 4.

  3. Cube the result: The second operation is to cube the result of the subtraction, which is (x−4)(x - 4). Cubing this expression means raising it to the power of 3, so we write (x−4)3(x - 4)^3.

  4. Express the function: Finally, we express the function f(x)f(x) as the result of these operations. Therefore, the algebraic formula for the function ff is:

    f(x)=(x−4)3f(x) = (x - 4)^3

This formula succinctly captures the verbal description, specifying that we first subtract 4 from the input xx and then cube the result.

Expanding and Simplifying (Optional)

While the formula f(x)=(x−4)3f(x) = (x - 4)^3 accurately represents the function, we can optionally expand and simplify it further. Expanding the expression (x−4)3(x - 4)^3 involves multiplying it out:

(x−4)3=(x−4)(x−4)(x−4)(x - 4)^3 = (x - 4)(x - 4)(x - 4)

We can first multiply (x−4)(x−4)(x - 4)(x - 4):

(x−4)(x−4)=x2−4x−4x+16=x2−8x+16(x - 4)(x - 4) = x^2 - 4x - 4x + 16 = x^2 - 8x + 16

Then, we multiply the result by (x−4)(x - 4):

(x2−8x+16)(x−4)=x3−4x2−8x2+32x+16x−64(x^2 - 8x + 16)(x - 4) = x^3 - 4x^2 - 8x^2 + 32x + 16x - 64

Combining like terms, we get:

x3−12x2+48x−64x^3 - 12x^2 + 48x - 64

Therefore, the expanded form of the function is:

f(x)=x3−12x2+48x−64f(x) = x^3 - 12x^2 + 48x - 64

While this expanded form is equivalent to the original formula f(x)=(x−4)3f(x) = (x - 4)^3, the original form is often preferred because it more clearly reflects the steps described in the verbal description: subtract 4, then cube the result. The expanded form obscures these individual operations.

Applying the Formula

To solidify our understanding, let's use the formula to evaluate the function for a specific input value. For example, let's find f(5)f(5):

f(5)=(5−4)3=(1)3=1f(5) = (5 - 4)^3 = (1)^3 = 1

So, when the input is 5, the output of the function is 1. Similarly, we can find f(2)f(2):

f(2)=(2−4)3=(−2)3=−8f(2) = (2 - 4)^3 = (-2)^3 = -8

When the input is 2, the output is -8. These examples demonstrate how the algebraic formula allows us to easily calculate the output of the function for any given input.

Key Takeaways

  • To translate a verbal description of a function into an algebraic formula, carefully identify the operations involved and their order.
  • Use variables to represent input values and construct the formula step-by-step, following the order of operations.
  • The formula f(x)=(x−4)3f(x) = (x - 4)^3 succinctly captures the verbal description "Subtract 4, then cube the result."
  • Expanding and simplifying the formula is optional but can sometimes obscure the original operations.
  • The algebraic formula allows us to easily calculate the output of the function for any given input.

By understanding this process, you can confidently translate verbal descriptions of functions into algebraic formulas, a crucial skill in various mathematical contexts.

Practice Problems

To further enhance your understanding, try translating the following verbal descriptions into algebraic formulas:

  1. Multiply by 2, then add 3.
  2. Add 1, then square the result.
  3. Divide by 5, then subtract 2.
  4. Take the square root, then multiply by 4.

Working through these practice problems will solidify your ability to convert verbal descriptions into algebraic expressions.

Expressing a Function Algebraically from a Verbal Description

Let's clarify the question: "A function ff has the following verbal description: 'Subtract 4, then cube the result.' Find a formula that expresses ff algebraically. $f(x) = $?"

The core of the question lies in translating the verbal description into a mathematical expression. We are given a function ff that performs two operations on an input: first, it subtracts 4 from the input, and then it cubes the result. The question asks us to represent this function using an algebraic formula, specifically in the form $f(x) = $?. This means we need to define how the function ff transforms an input xx based on the given description.

Breaking Down the Verbal Description

To formulate the algebraic expression, we need to carefully dissect the verbal description. There are two key operations described:

  1. Subtraction: The first operation is to subtract 4 from the input.
  2. Cubing: The second operation is to cube the result obtained after the subtraction.

The order of these operations is critical. We must perform the subtraction before cubing the result. This sequential execution is what we need to capture in our algebraic expression.

Constructing the Algebraic Formula

Now, let's translate these operations into algebraic symbols. Let's consider the input to our function as 'x'.

  1. Subtracting 4: When we subtract 4 from the input xx, we get the expression (x−4)(x - 4). This represents the first operation.

  2. Cubing the Result: Next, we need to cube the entire result obtained in the previous step. Cubing a quantity means raising it to the power of 3. So, we take (x−4)(x - 4) and raise it to the power of 3, which gives us (x−4)3(x - 4)^3.

  3. Expressing the Function: Finally, we express the function f(x)f(x) as the result of these operations performed in sequence. Therefore, the algebraic formula for the function ff is:

    f(x)=(x−4)3f(x) = (x - 4)^3

This formula accurately represents the verbal description, indicating that we first subtract 4 from xx and then cube the resulting difference. This is the algebraic expression we were seeking.

Evaluating the Formula

To ensure our formula is correct, we can test it with a few example inputs. Let's try a couple of values for xx:

  • If x=5x = 5, then f(5)=(5−4)3=(1)3=1f(5) = (5 - 4)^3 = (1)^3 = 1.
  • If x=0x = 0, then f(0)=(0−4)3=(−4)3=−64f(0) = (0 - 4)^3 = (-4)^3 = -64.

These evaluations demonstrate how the formula transforms the input based on the described operations. The formula provides a clear and concise way to compute the output of the function for any given input xx.

Importance of Order of Operations

The order of operations is crucial in this problem. If we were to cube first and then subtract 4, the formula would be entirely different and incorrect. For example, if we misinterpreted the description and wrote f(x)=x3−4f(x) = x^3 - 4, this would mean we cube xx first and then subtract 4, which is not what the verbal description specifies. The parentheses in (x−4)3(x - 4)^3 ensure that subtraction occurs before cubing, aligning perfectly with the description.

Key Points to Remember

  • When translating verbal descriptions to algebraic formulas, identify each operation and its order.
  • Use parentheses to group operations that need to be performed together.
  • Test the formula with example inputs to verify its correctness.
  • Pay close attention to the order of operations to ensure the formula accurately represents the verbal description.

In summary, the algebraic formula f(x)=(x−4)3f(x) = (x - 4)^3 correctly expresses the function described verbally as "Subtract 4, then cube the result." This question underscores the importance of translating verbal descriptions accurately into mathematical notation, a fundamental skill in algebra.

Additional Examples and Practice

To further solidify understanding, consider the following examples of translating verbal descriptions into algebraic formulas:

  1. "Multiply by 3, then add 2": This translates to f(x)=3x+2f(x) = 3x + 2.
  2. "Add 5, then divide by 2": This translates to f(x)=(x+5)/2f(x) = (x + 5) / 2.
  3. "Square the input, then subtract 1": This translates to f(x)=x2−1f(x) = x^2 - 1.

These examples reinforce the process of breaking down the description and representing each operation algebraically. Practice with similar problems will enhance your ability to translate verbal descriptions into algebraic formulas confidently.

Function Formula from Verbal Description - A Step-by-Step Guide