Question

The value of the function \(f(x)=3 x-7\) at 5 is _____

Short answer

8

Step-by-step solution

Identify the given function

The given function is \(f(x) = 3x - 7\).

Substitute the given value into the function

We need to find the value of the function at \(x = 5\). Substitute \(x = 5\) into the function: \(f(5) = 3(5) - 7\).

Simplify the expression

Calculate the expression: \(3(5) - 7 = 15 - 7 = 8\).

Write the final answer

The value of the function \(f(x) = 3x - 7\) at \(x = 5\) is \(8\).

Key concepts

Linear Functions

A linear function is one of the simplest types of functions you can encounter in algebra. It generally has the form:

• Standard form: \[ f(x) = mx + b \]

In this form, m represents the slope of the line, and b represents the y-intercept.
For the given exercise, our linear function is: \[ f(x) = 3x - 7 \]. Here:

• Slope (m): 3 (This tells us how steep the line is.)
• Intercept (b): -7 (This tells us where the line crosses the y-axis.)

Understanding these functions is crucial because they are foundational in algebra and are widely used in real-life scenarios, like in economics for calculating costs and profits.

Substitution Method

The substitution method helps solve equations by replacing variables with known values. For the given function: \[ f(x) = 3x - 7 \]To find its value at a specific point, let’s say x=5, follow these steps:

• Step 1: Identify the given function value. Here, it’s x = 5.
• Step 2: Substitute x with 5 in the function. This gives us:
  \[ f(5) = 3(5) - 7 \].

Simply put, we replace 'x' with 5 wherever 'x' appears in the function.This step is very crucial as it leads to the next and final step, where we perform the arithmetic to find the required value of the function.

Simplifying Expressions

Simplifying expressions involves performing basic arithmetic operations to get a simpler form of an equation. For our problem: We have: \[ f(5) = 3(5) - 7 \].First, we perform the multiplication:

• \[ 3(5) = 15 \]

Next, we subtract 7 from 15:

• \[ 15 - 7 = 8 \]

So, the simplified form of the expression is 8.Understanding how to simplify expressions is essential because it makes complex problems more manageable. It also prepares you for more advanced topics in mathematics.