Problem 37 Given the following functions, f... [FREE SOLUTION]

91影视

Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 3: Problem 37

Given the following functions, find the function values. $f(x)=5 x-9,$ find $x$ when $f(x)=1$.

Short Answer

Expert verified

x = 2.

Step by step solution

Set up the Equation

We are given that $f(x) = 5x - 9$ and are asked to find $x$ when $f(x) = 1$. So, set up the equation $5x - 9 = 1$.

Isolate the Variable Term

Add 9 to both sides of the equation to move the constant term to the other side: $5x - 9 + 9 = 1 + 9$, which simplifies to $5x = 10$.

Solve for x

Divide both sides of the equation by 5 to solve for $x$: $x = \frac{10}{5}$.

Simplify the Result

Simplify the simplified fraction to find $x = 2$.

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Key Concepts

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

Functions

In mathematics, functions play a crucial role, acting as a special kind of relationship between two sets: the domain and the range. A function, typically denoted as $ f(x) $, describes how each input (or $ x $ value) from the domain is associated with exactly one output (or $ f(x) $ value) in the range.

Expression: The example, $ f(x) = 5x - 9 $, expresses a linear function where $ 5x - 9 $ defines the rule of assigning outputs to inputs.
Input and Output: Here, $ x $ is the input variable, and $ f(x) $ gives the output after applying the function rule.
Finding Values: To determine $ x $ when $ f(x) = 1 $, we solve the equation so that the rule $ 5x - 9 $ produces the output 1.

Understanding functions is fundamental to grasp the behaviors of different mathematical relationships. They provide a framework to model real-world problems and predict outcomes based on certain inputs.

Linear equations

Linear equations are one of the building blocks of algebra. They are equations of the first degree, which means they are polynomials with a maximum power of one.

Form: A typical linear equation format is $ ax + b = c $, where $ a $, $ b $, and $ c $ are constants, and $ x $ is the variable.
Example: In our exercise, the equation $ 5x - 9 = 1 $ is a linear equation derived from setting the function $ f(x) $ equal to a specific value (here, 1).
Graph: Graphically, linear equations represent straight lines with a consistent slope, determined by the coefficient of $ x $ (here, 5), and a y-intercept where the line crosses the y-axis ($ -9 $ in our function).

Linear equations are widely used to solve various mathematical and practical problems, allowing us to calculate and predict straightforward scenarios with two variables.

Solving equations

Solving equations involves finding the value(s) of the variable(s) that make the equation true. The process requires understanding and manipulating mathematical operations to isolate the variable.

Step 1: Begin by organizing and rewriting the equation, as in our example: $ 5x - 9 = 1 $.
Step 2: Use inverse operations to isolate the variable. Add 9 to both sides to cancel the $-9$, resulting in $ 5x = 10 $.
Step 3: Simplify the equation by dividing both sides by 5, yielding $ x = \frac{10}{5} $.
Step 4: Further simplify, if possible, to find the final solution: $ x = 2 $.

The key to solving any equation is to perform the same operations on both sides, keeping the equation balanced. Mastering this concept allows us to tackle increasingly complex algebraic equations with confidence.

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91影视

Given the following functions, find the function values. \(f(x)=5 x-9,\) find \(x\) when \(f(x)=1\).

Short Answer

Step by step solution

Set up the Equation

Isolate the Variable Term

Solve for x

Simplify the Result

Key Concepts

Functions

Linear equations

Solving equations

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