Question

Challenge Problem If \(f\left(\frac{x+4}{5 x-4}\right)=3 x^{2}-2,\) find \(f(1)\)

Short answer

f(1) = 10

Step-by-step solution

- Set up the equation

First, recognize the given function: \(f\left(\frac{x+4}{5x-4}\right) = 3x^2 - 2\)

- Set the argument equal to 1

We need to find the value of \(x\) such that the argument of the function is 1. Set the inner part of the function to 1: \(\frac{x+4}{5x-4} = 1\)

- Solve for x

Solve the equation \(\frac{x+4}{5x-4} = 1\): \((x + 4) = (5x - 4)\)Solve for \(x\):\(x + 4 = 5x - 4\)\(4 + 4 = 5x - x\)\(8 = 4x\)\(x = 2\)

- Substitute x into the function

With \(x = 2\), substitute it back into the expression for the function: \(f\left(\frac{2+4}{5(2)-4}\right) = 3(2)^2 - 2\)\(\frac{6}{10-4} = 1\), so the function is \(3(4) - 2 = 12 - 2 = 10\)

- Conclusion

Thus, \(f(1) = 10\)

Key concepts

Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions containing variables. In our problem, we dealt with the specific equation: \(\frac{x + 4}{5x - 4} = 1\).
To solve this, we first recognized that algebraic manipulation would be needed.
By equating \(\frac{x + 4}{5x - 4}\) with 1, we created a base for solving it by isolating variables on one side.
Through basic algebraic steps—such as combining like terms and simplifying—we determined the correct value for 'x'.

Function Evaluation

Function evaluation involves determining the output of a function for a specific input.
We are given the function in the form: \(f\biggl(\frac{x + 4}{5x - 4}\biggr) = 3x^2 - 2\).
Our task is to find \(f(1)\). To do this, we needed to determine the value of \(\frac{x + 4}{5x - 4}\) such that it equals 1.
When \(x = 2\), the function simplifies to \(f(1) = 3(2)^2 - 2\) resulting in \(10\).
This demonstrates how evaluating a function requires substituting the correct values and simplifying appropriately according to given conditions.

Solving for Variables

Solving for variables is an essential skill in algebra, where the goal is to determine the unknowns.
In the provided problem, we found 'x' by setting the equation \(\frac{x + 4}{5x - 4} = 1\) and solving it step-by-step:

• We first acknowledged that \(\frac{x + 4}{5x - 4} = 1\) implies \(x + 4 = 5x - 4\).
• Next, we combined like terms and simplified: \(4 + 4 = 5x - x \rightarrow 8 = 4x\) and finally, \(x = 2\).

Substituting \(x\) back into the function provided us with the needed variable value to solve for the final function output.