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Chapter 2: Problem 96

Given the linear function defined by \(f(x)=2 x-5\), simplify the following. $$f(0)-f(7)$$

Short Answer

Expert verified
The simplified expression is -14.

Step by step solution

01

Evaluate f(0)

To find the value of the function at 0, plug 0 into the linear equation. \[ f(0) = 2(0) - 5 \]Simplifying this gives:\[ f(0) = -5 \]
02

Evaluate f(7)

Substitute 7 into the function to find the value:\[ f(7) = 2(7) - 5 \]Calculating this, we get:\[ f(7) = 14 - 5 = 9 \]
03

Calculate f(0) - f(7)

Now subtract the result of \( f(7) \) from \( f(0) \):\[ f(0) - f(7) = -5 - 9 \]Simplify the expression to:\[ -5 - 9 = -14 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Evaluation
A linear function, like the one given as \( f(x) = 2x - 5 \), can take any number \( x \) and applies a set of rules to give us a new number, \( f(x) \). This is known as evaluating the function. To do this:
  • Substitute the given number into the function wherever you see \( x \).
  • Perform any necessary arithmetic operations following this substitution.
This step allows us to see exactly how the function behaves at any given \( x \) value. In our exercise, we evaluated the function with specific values, first at \( x = 0 \) and then at \( x = 7 \).
By substituting these values, we obtained \( f(0) = -5 \) and \( f(7) = 9 \). Evaluating the function is vital as it helps us understand how changes in \( x \) affect the overall output of \( f(x) \).
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form. This is crucial in solving mathematical problems efficiently. During simplification, you perform arithmetic operations like addition, subtraction, multiplication, and division. For our specific problem, we simplified the expressions:
  • \( f(0) = 2(0) - 5 = -5 \)
  • \( f(7) = 2(7) - 5 = 9 \)
Each was reduced to a single number after performing the necessary operations, making it easier to work with.
Finally, we calculated \( f(0) - f(7) \) and simplified it as follows:
  • \( -5 - 9 = -14 \)
Simplifying not only makes expressions easier to handle but also helps avoid potential mistakes in further calculations.
Step by Step Solutions
Taking a step-by-step approach makes solving mathematical problems more manageable and less overwhelming. By breaking down each part of the exercise, we can focus on one operation at a time and ensure we understand each concept before moving on. For the linear function \( f(x) = 2x - 5 \), we followed these steps:
  • First, we found \( f(0) \): Substituted 0 in place of \( x \) and performed the arithmetic to get \( f(0) = -5 \).
  • Next, we evaluated \( f(7) \): By substituting 7 for \( x \), we calculated \( f(7) = 9 \).
  • Finally, we subtracted: Took the results from the two previous calculations to find \( f(0) - f(7) = -14 \).
By following this orderly sequence, each step builds on the other, leading to the ultimate solution while keeping errors at bay and reinforcing understanding.

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