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
4 years, 6 months ago

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