Question

. If $f(x)=2x-3$, what is the value of $f(x)$ when $x$ is $-5$ ? (A) $-13$ (B) $-7$ (C) 7 (D) 11

Short answer

The value of $f(x)$ when $x$ is $-5$ is -13. The correct answer is (A) $-13$.

Step-by-step solution

Write down the given function

The given function is $f(x)=2x-3$.

Substitute x with -5

We will replace $x$ with $-5$ in the function: $f(-5)=2(-5)-3$.

Simplify the expression

Let's simplify the expression: $f(-5)=2(-5)-3=-10-3=-13$.

State the answer

The value of $f(x)$ when $x$ is $-5$ is -13. So, the correct answer is (A) $-13$.

Key concepts

Understanding Functions

In mathematics, a function is a relationship between two variables, typically denoted as $f(x)$ where $x$ is the independent variable and $f(x)$ is the output or dependent variable. Functions describe how each input $x$ corresponds to exactly one output $f(x)$. This concept is key to many areas of math, including algebra and calculus.

Functions are often written in the form of an equation, such as $f(x)=2x-3$. In this equation:

• The function rule or formula shows us the operation we perform on the input $x$ to get the output $f(x)$.
• "$2x-3$" is the expression that allows us to calculate the output depending on the input $x$.
• We can evaluate a function by substituting a number for $x$ and solving for $f(x)$.

Understanding how to work with functions allows you to analyze relationships and changes between quantities effectively.

Applying the Substitution Method

The substitution method is a common and efficient technique for solving algebraic expressions or functions. The basic idea is to replace a variable with a given value to find the outcome of an expression. This method is particularly useful in evaluating functions, as seen in the example of $f(x)=2x-3$.

To use the substitution method:

• Identify the variable you need to substitute, in this case, $x$.
• Take the given value of $x$, which is $-5$ in the exercise, and replace $x$ in the function.
• For $f(x)=2(-5)-3$, you perform the operation defined by the function formula after substituting $-5$ for $x$.

This process highlights how substitution can simplify complex expressions and solve problems, providing a clear pathway to finding solutions.

Solving Linear Equations

Linear equations are mathematical expressions in which the highest power of the variable is one. They form straight lines when graphed, hence the term 'linear'. An example of a linear equation is $y=2x-3$, similar to the function in our exercise.

Key points about linear equations:

• They can be written in the standard form $ax+b=0$, but also appear in forms that solve for $y$ or $f(x)$ like $y=mx+c$.
• The coefficients and constants in the equation determine the line's slope and position on a graph.
• Solving linear equations involves finding the value of the variable that makes the equation true.

In the context of the exercise, we substituted $-5$ into a linear function and solved for $f(x)$, illustrating a vital use of linear equations in real-world problem-solving.