Question

If \(f(x)=\frac{3 x+8}{2 x-A}\) and \(f(0)=2,\) what is the value of \(A ?\)

Short answer

The value of \(A\) is \(-4\).

Step-by-step solution

- Substitute the Given Value in the Function

Start by substituting the given value of the function, which is when \(x = 0\), into the function \(f(x) = \frac{3x + 8}{2x - A}\). This will help in solving for \(A\).So, \(f(0) = \frac{3(0) + 8}{2(0) - A} = 2\)

- Simplify the Equation

Simplify the equation \(\frac{8}{-A} = 2\). This simplifies to \(8 = 2(-A)\).

- Solve for \(A\)

Divide both sides of the equation by 2 to isolate \(A\). \(8 = -2A \rightarrow -A = 4 \rightarrow A = -4\).

Key concepts

Function Evaluation

Function evaluation is the process of finding the output of a function for a given input. In our exercise, we need to evaluate the function when the input is 0. The given rational function is:
\( f(x) = \frac{3x + 8}{2x - A} \)
To evaluate the function, we substitute 0 for x:
\( f(0) = \frac{3(0) + 8}{2(0) - A} = \frac{8}{-A} \)
This simplifies our equation. Properly substituting values makes function evaluation simple and straightforward.

Solving Equations

In solving equations, our goal is to isolate the variable we need to find. In the exercise, after evaluating the function at 0, we have:
\( f(0) = \frac{8}{-A} = 2 \)
To find the value of A, we need to solve this equation.
First, we remove the fraction by multiplying both sides of the equation by -A:
\[ 8 = 2(-A) \]
Next, we simplify the equation to isolate A:
\[ 8 = -2A \]
Dividing both sides by -2 gives:
\[ A = -4 \]
Solving equations involves systematic steps to isolate the variable. Remember to simplify step by step and do the same operations on both sides of the equation.

Finding Parameters in Functions

Finding parameters in functions is about determining unknown values within function definitions. In our exercise, we are given the rational function:
\( f(x) = \frac{3x + 8}{2x - A} \)
and the condition that \ f(0) = 2\. Our goal is to find A that satisfies this condition.
Starting with \[ f(0) = \frac{8}{-A} = 2 \]
and solving the equation helps us determine that:
\[ 8 = -2A \]
Thus,
\[ A = -4 \]
By understanding the structure of the function and using the given conditions, we can find unknown parameters. This process is essential for accurately defining functions and ensuring they meet specified criteria.