Problem 28 - Precalculus

Question

Use the given function \(f\) to find \(f(0)\) and solve \(f(x)=0\) $$f(x)=3-\frac{2}{5} x$$

Step-by-Step Solution

Answer

\(f(0) = 3\) and \(x = 7.5\) for \(f(x) = 0\).

Step 1: Substitute for f(0)

To find the value of \(f(0)\), substitute \(x = 0\) into the function \(f(x) = 3 - \frac{2}{5}x\). This gives \(f(0) = 3 - \frac{2}{5}(0) = 3\).

Step 2: Set the Function Equal to Zero

To solve the equation \(f(x) = 0\), set the function equal to zero: \(0 = 3 - \frac{2}{5}x\).

Step 3: Isolate x

Rearrange the equation to isolate \(x\): \(\frac{2}{5}x = 3\) by subtracting 3 from both sides and then multiplying both sides by \(\frac{5}{2}\) to solve for \(x\): \(x = \frac{5}{2} \times 3\).

Step 4: Calculate x

Calculate the value of \(x\): \(x = \frac{5}{2} \times 3\), which simplifies to \(x = \frac{15}{2}\) or \(x = 7.5\).

Key Concepts

Function Evaluation

When evaluating a function, we essentially plug a given input into a function to find its corresponding output. A function, like the one given here, is an equation that relates an input value to an output value. In the exercise, we have the function \( f(x) = 3 - \frac{2}{5}x \). To evaluate \( f(0) \), we replace the variable \( x \) with 0, yielding:

\[ f(0) = 3 - \frac{2}{5}(0) = 3 \]

This tells us that the output value when the input \( x = 0 \) is the constant 3. Function evaluation is crucial because it allows us to understand how changes in the input value affect the output.

Solving Equations

Solving an equation means finding the value of the variable that makes the equation true. In our exercise, we need to solve \( f(x) = 0 \). This means we're looking for the \( x \) value that results in the output of the function being zero.

Starting with the equation:

\[ 0 = 3 - \frac{2}{5}x \]

we aim to find the value of \( x \) that makes this true. We do this by setting the function equal to zero and solving for \( x \). This gives us a clearer picture of where the graph of the function intersects the x-axis.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations and expressions to isolate variables or simplify expressions. In the solution to the given problem, we performed several algebraic manipulations to solve for \( x \).

Initially, we have:

\[ \frac{2}{5}x = 3 \]

By multiplying both sides by \( \frac{5}{2} \), we cancel out the fraction on the left, leaving:

\[ x = \frac{5}{2} \times 3 \]

This simplifies to:

\[ x = \frac{15}{2} \] or \( x = 7.5 \).

This step of manipulating the equation is essential when solving formulas and helps us understand how different components of an equation relate to each other.