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# Evaluate Expressions

## 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}

Note by Arron Kau
2 years, 12 months ago