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Chapter 3: Problem 53

If \(f(x)=8 x-9\), evaluate \(f(3)\).

Short Answer

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Step by step solution

01

Understand the Function

The function given is \( f(x) = 8x - 9 \). To evaluate \( f(3) \), we need to substitute 3 for \( x \) in the function.
02

Substitute the Value

Substitute 3 into the function where \( x \) is: \( f(3) = 8(3) - 9 \).
03

Simplify the Expression

Calculate the value by first multiplying then subtracting: \( 8 \times 3 = 24 \), so \( f(3) = 24 - 9 \).
04

Find the Result

Perform the subtraction to get the final result: \( f(3) = 24 - 9 = 15 \). Thus, \( f(3) = 15 \).

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Key Concepts

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

Substitution
Substitution is the core process we use to evaluate a function for a specific value. In the context of functions, substitution means replacing a variable with a given number.
For example, if you have the function \( f(x) = 8x - 9 \), and you need to find \( f(3) \), you substitute 3 for \( x \).
So, it looks like this: \( f(3) = 8(3) - 9 \).
This initial step sets the stage for the entire process of evaluating the function.
Simplification
Simplification follows substitution in function evaluation.
After you have substituted the value into the function, you must perform arithmetic operations to simplify the expression.
In our example, the function after substitution is \( f(3) = 8(3) - 9 \).
We simplify it in two steps:
  • First, multiply: \( 8 \times 3 = 24 \)
  • Second, subtract: \( 24 - 9 = 15 \)
By simplifying the expression step-by-step, we arrive at the answer \( f(3) = 15 \). Simplification helps in getting a clear numerical result from the substituted values.
Function Notation
Function notation is a way to represent functions in a standardized form.
It's commonly written as \( f(x) \), where \( f \) denotes the function and \( x \) is the variable.
This notation helps to easily substitute and simplify values within different functions.
For instance, in \( f(x) = 8x - 9 \), \( f(x) \) indicates a function of \( x \).
If we need to evaluate it for \( x = 3 \), we write \( f(3) \), meaning we replace \( x \) with 3.
Function notation makes it easier to handle functions, and it plays a key role in calculus and algebra by providing a clear and concise way to denote function operations.

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