Chapter 12: Problem 9
If \(?(x)=\frac{x^{2}+3}{x-1}\) and \(3 r-4=5,\) what is the value of ?(r) ?$
Short Answer
Expert verified
\(?(3) = 6\)
Step by step solution
01
- 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, \(r = 3\).
02
- 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 \(?(3)\).
03
- 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: \[?(3) = \frac{9 + 3}{2} = \frac{12}{2} = 6\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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
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:
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
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:
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
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:
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