## Definition

To **evaluate** expressions, you substitute specific values for variables. For example, to evaluate the expression \(\sqrt{x+2}\) for \(x=2\), you substitute \(2\) in for \(x\) to get \(\sqrt{2+2} = 2\).

## Technique

Evaluating functions works similarly. For example, given a function \(f(x) = \sqrt{x+2}\), then \( f(2) = \sqrt{2+2}=2 \).

For expressions and functions with multiple variables, the procedure is still the same. For example, given that \(f(x,y)=x^2+2y-1\), \(f(-4,1)=(-4)^2+2(1)-1=17\).

Sometimes you are given the value of an expression but not of the variables. In these cases, you may be asked to solve for variable values.

Here is a basic one-variable example:

## What value of \(x\) satisfies \(3x+7=46\)?

Solving for \(x\), we have: \[\begin{align}

3x + 7 &= 46\\

3x &= 46-7\\

3x &= 39\\

x &= 13\quad_\square\

\end{align}\]

Here is a two-variable example:

## For the function \(f(x,y)=x^2+y\), how many ordered pairs of integers satisfy \(f(x,y)=1\) if \(-10 \leq x,y \leq 10\)?

\(f(x,y)=x^2+y=1\) can be rewritten as \(y = 1 - x^2\), a parabola. Given the constraints on \(x\) and \(y\), our ordered pairs of integers are \((0,1)\), \((\pm 1,0)\), \((\pm 2,-3)\), and \((\pm 3,-8)\) for a total of \(7\) pairs. \(_\square\)

## Application and Extensions

## What value of \(n\) satisfies \((n+2)^2 - n^2 = 16\)?

We can solve for \(n\) after multiplying out and combining terms. \[\begin{align}

n^2+4n+4 - n^2 &= 16 \\

4n+4 &=16 \\

4(n+1) &=16 \\

n+1 &= 4\\

n &= 3 \quad_\square\

\end{align}\]

## Given that \(f(x)=\frac{6x^2 - 11x - 7}{2x+1}\), what is \(f(7)\)?

While we can substitute in \(7\) in for \(x\) right away, the function is easier to evaluate if we first simplify it via factoring. \[\begin{align}

f(x) &= \frac{6x^2 - 11x - 7}{2x+1} \\

&= \frac{(2x+1)(3x-7)}{2x+1} \\

&= 3x-7 \\

f(7)&=3(7)-7 \\

f(7)&=14 \quad_\square\

\end{align}\]

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