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 $\sqrt{2+2} = 2$.

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: $\begin{aligned} 3x + 7 &= 46\\ 3x &= 46-7\\ 3x &= 39\\ x &= 13\quad_\square\ \end{aligned}$

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. $_\square$

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

We can solve for $n$ after multiplying out and combining terms. $\begin{aligned} n^2+4n+4 - n^2 &= 16 \\ 4n+4 &=16 \\ 4(n+1) &=16 \\ n+1 &= 4\\ n &= 3 \quad_\square\ \end{aligned}$

## 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. $\begin{aligned} f(x) &= \frac{6x^2 - 11x - 7}{2x+1} \\ &= \frac{(2x+1)(3x-7)}{2x+1} \\ &= 3x-7 \\ f(7)&=3(7)-7 \\ f(7)&=14 \quad_\square\ \end{aligned}$

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