Evaluate Expressions

Definition

To evaluate expressions, you substitute specific values for variables. For example, to evaluate the expression x+2\sqrt{x+2} for x=2x=2, you substitute 22 in for xx to get 2+2=2\sqrt{2+2} = 2.

Technique

Evaluating functions works similarly. For example, given a function f(x)=x+2f(x) = \sqrt{x+2}, then f(2)=2+2=2 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)=x2+2y1f(x,y)=x^2+2y-1, f(4,1)=(4)2+2(1)1=17f(-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 xx satisfies 3x+7=463x+7=46?

Solving for xx, we have: 3x+7=463x=4673x=39x=13 \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)=x2+yf(x,y)=x^2+y, how many ordered pairs of integers satisfy f(x,y)=1f(x,y)=1 if 10x,y10-10 \leq x,y \leq 10?

f(x,y)=x2+y=1f(x,y)=x^2+y=1 can be rewritten as y=1x2y = 1 - x^2, a parabola. Given the constraints on xx and yy, our ordered pairs of integers are (0,1)(0,1), (±1,0)(\pm 1,0), (±2,3)(\pm 2,-3), and (±3,8)(\pm 3,-8) for a total of 77 pairs. _\square

Application and Extensions

What value of nn satisfies (n+2)2n2=16(n+2)^2 - n^2 = 16?

We can solve for nn after multiplying out and combining terms. n2+4n+4n2=164n+4=164(n+1)=16n+1=4n=3 \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)=6x211x72x+1f(x)=\frac{6x^2 - 11x - 7}{2x+1}, what is f(7)f(7)?

While we can substitute in 77 in for xx right away, the function is easier to evaluate if we first simplify it via factoring. f(x)=6x211x72x+1=(2x+1)(3x7)2x+1=3x7f(7)=3(7)7f(7)=14 \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}

Note by Arron Kau
5 years, 8 months ago

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