Question

If \(?(x)=\frac{x^{2}+3}{x-1}\) and \(3 r-4=5,\) what is the value of ?(r) ?$

Short answer

\(?(3) = 6\)

Step-by-step solution

- Solve for r

Given the equation \(3r - 4 = 5\), solve for \(r\) by isolating \(r\). Start by adding 4 to both sides of the equation: \[3r - 4 + 4 = 5 + 4\] This results in \(3r = 9\). Next, divide both sides by 3: \[r = \frac{9}{3}\]. Thus, \(r = 3\).

- Substitute r into the function

Substitute the value of \(r\) into the function \(?(x)=\frac{x^{2}+3}{x-1}\). With \(r = 3\), we need to find \(?(3)\).

- Compute the value of \(?(3)\)

Substitute \(x = 3\) into the function: \[?(3) = \frac{3^{2} + 3}{3 - 1}\] Simplify the expression inside the numerator and the denominator: \[?(3) = \frac{9 + 3}{2} = \frac{12}{2} = 6\]

Key concepts

Algebra

Algebra is the branch of mathematics that deals with symbols and the rules governing the manipulation of these symbols. In this problem, we use algebra to solve for the variable r in the equation 3r - 4 = 5. Start by isolating r, which involves performing operations to both sides of the equation to get r alone.
To isolate r:

• Add 4 to both sides: 3r - 4 + 4 = 5 + 4
• Simplify to get 3r = 9
• Divide both sides by 3 to get r = 3

Basic algebraic techniques like addition, subtraction, multiplication, and division are critical in solving such equations. Understanding these fundamental operations can help you solve more complex algebraic problems.

Function Evaluation

Function evaluation is the process of finding the output of a function given an input. In this problem, we have the function ?(x)= \[ \frac{x^{2}+3}{x-1} \]. After finding the value of r, which is 3, we need to evaluate the function at this input.
Steps to evaluate the function ?(x) at x = 3:

• Substitute x with 3 in the function: ?(3) = \[ \frac{3^{2}+3}{3-1} \]
• Simplify the expression inside the numerator and the denominator: ?(3) = \[ \frac{9+3}{2} \]
• Carry out the division: ?(3) = 6

Evaluating functions is straightforward—simply substitute the given input and simplify to find the corresponding output.

Solving Equations

Solving equations involves finding the value(s) of the variable(s) that make the equation true. The equation from the problem is 3r - 4 = 5. The goal is to find r.
Steps to solve the equation:

• Add 4 to both sides to cancel out the -4: 3r - 4 + 4 = 5 + 4
• This simplifies to 3r = 9
• Next, divide both sides by 3: r = \[ \frac{9}{3} \]
• Simplify to find r = 3

The key is to perform equivalent operations on both sides of the equation to isolate the variable. Equate steps are vital for ensuring equations are balanced throughout the solution process.

Substitution Method

The substitution method involves replacing one variable with its equivalent value. In this problem, after solving for r, we substitute it into the function.
Steps for using the substitution method:

• First, solve the initial equation to find the value of r: r = 3
• Next, substitute r into the function ?(x) replacing x: ?(3) = \[ \frac{3^{2}+3}{3-1} \]
• Simplify to find the result: ?(3) = 6

This method is useful because it allows you to break down complex problems into more manageable parts by focusing on one variable at a time. After isolating and solving for one variable, substitute it back into the equation or function to find the final result.