Question

For \(f(x)=\frac{2}{5} x+\frac{4}{5}\), find \(f(-7)\) A. -14 B. -10 C. -2 D. 5

Short answer

C. -2

Step-by-step solution

- Understand the Function

The function given is \( f(x) = \frac{2}{5} x + \frac{4}{5} \). This is a linear function of the form \( f(x) = mx + b \) where \( m = \frac{2}{5} \) and \( b = \frac{4}{5} \).

- Substituting the Given Value

We need to find \( f(-7) \). Substitute \( x = -7 \) into the function: \( f(-7) = \frac{2}{5}(-7) + \frac{4}{5} \).

- Simplify Inside the Function

Calculate \( \frac{2}{5}(-7) = \frac{2 \times -7}{5} = \frac{-14}{5} \). So now we have: \[ f(-7) = \frac{-14}{5} + \frac{4}{5} \]

- Combine the Fractions

Combine the fractions on the right side of the equation: \[ f(-7) = \frac{-14 + 4}{5} = \frac{-10}{5} \]

- Simplify the Result

Simplify the fraction: \( \frac{-10}{5} = -2 \). Therefore, \( f(-7) = -2 \).

Key concepts

Evaluating Functions

Evaluating functions is essentially the process of finding the value of the function for a particular input. In the given exercise, the function is defined as: \[ f(x) = \frac{2}{5} x + \frac{4}{5} \]To evaluate the function at a specific value, like \(x = -7\), we substitute the given value into the function and calculate the result. This involves basic arithmetic operations and ensures we follow the order of operations (PEMDAS/BODMAS).

• First, replace \(x\) with the given value.
• Then, perform the arithmetic operations step-by-step.
• Finally, simplify the result if necessary.

Evaluating functions is a crucial skill in algebra, as it helps us understand the behavior of the function at different points.

Substitution in Algebra

Substitution in algebra involves replacing a variable with a specific value. It is a fundamental technique that helps in solving equations or simplifying expressions. For the given function:\[ f(x) = \frac{2}{5} x + \frac{4}{5} \]To find \( f(-7) \), we substitute \( x = -7 \) into the function: \[ f(-7) = \frac{2}{5}(-7) + \frac{4}{5} \]Here are the steps for substitution:

• Identify the variable to substitute.
• Replace the variable with the given value.
• Simplify the resulting expression.

When substituting, it's important to carefully carry out multiplication and addition operations to ensure accuracy. This technique is widely used in many areas of mathematics, including evaluation of functions, solving equations, and simplifying expressions.

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form. This makes it easier to understand and work with. In the exercise, after substituting \( x = -7 \) into the function, we are left with:\[ f(-7) = \frac{-14}{5} + \frac{4}{5} \]To simplify this, we combine the fractions by adding the numerators and maintaining a common denominator:\[ f(-7) = \frac{-14 + 4}{5} = \frac{-10}{5} \]The final step involves simplifying \( \frac{-10}{5} \):

• Divide the numerator by the denominator.
• In this case, \( \frac{-10}{5} = -2 \).

Simplifying fractions ensures that the result is in the most reduced form, making it clear and easy to interpret. This step is essential in various mathematical procedures, including evaluating expressions and solving equations.