Evaluating Radical and Rational Functions

Function Definition: A radical function involving a principal square root is defined as: $g(m) = \sqrt{m - 4}$
Evaluation Task: Determine the value of $g(5)$ by evaluating the function $g(m)$ at the input $m = 5$ .
Step-by-Step Evaluation Process:
- Step 1: Input Substitution Substitute the value $m = 5$ into the function in place of the variable $m$ : $g(5) = \sqrt{5 - 4}$
- Step 2: Radicand Simplification Perform the subtraction inside the radical sign (the radicand): $5 - 4 = 1$ $g(5) = \sqrt{1}$
- Step 3: Principal Square Root Calculation Evaluate the principal square root of $1$ : $\sqrt{1} = 1$
- Final Output: $g(5) = 1$

Function Definition: A rational function composed of linear expressions in the numerator and denominator is defined as: $f(x) = \frac{6x - 10}{5x + 4}$
Evaluation Task: Determine the value of $f(2)$ by evaluating the function $f(x)$ at the input $x = 2$ .
Step-by-Step Evaluation Process:
- Step 1: Input Substitution Substitute $x = 2$ into all instances of $x$ in both the numerator and denominator: $f(2) = \frac{6(2) - 10}{5(2) + 4}$
- Step 2: Numerator Evaluation Multiply $6$ by $2$ and then subtract $10$ : $6(2) = 12$ $12 - 10 = 2$
- Step 3: Denominator Evaluation Multiply $5$ by $2$ and then add $4$ : $5(2) = 10$ $10 + 4 = 14$
- Step 4: Form the Unsimplified Fraction Combine the evaluated numerator and denominator: $f(2) = \frac{2}{14}$
- Step 5: Simplification to Lowest Terms Reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $2$ : $\frac{2 \div 2}{14 \div 2} = \frac{1}{7}$
- Final Output: $f(2) = \frac{1}{7}$

Key Evaluation Rules:
- Radical Functions: Always complete arithmetic operations inside the square root prior to taking the principal root.
- Rational Functions: Separately evaluate the polynomial in the numerator and the polynomial in the denominator before performing any simplification or division.