Introduction to Functions

Function notation, domain, range, and evaluation

What is a Function?

A function is a relation where each input has exactly one output.

$f(x) = 2x + 3$

To find $f(5)$ when $f(x) = 2x + 3$ : $f(5) = 2(5) + 3 = 13$

A graph represents a function if any vertical line intersects it at most once.

If $f(x) = 2x - 5$ , find $f(3)$

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To find $f(3)$ , substitute $x = 3$ into the function:

$f(x) = 2x - 5$ $f(3) = 2(3) - 5$ $f(3) = 6 - 5$ $f(3) = 1$

Answer: $f(3) = 1$

Given $g(x) = x^2 - 3x + 1$ , find $g(-2)$

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Substitute $x = -2$ into the function:

$g(x) = x^2 - 3x + 1$ $g(-2) = (-2)^2 - 3(-2) + 1$ $g(-2) = 4 + 6 + 1$ $g(-2) = 11$

Answer: $g(-2) = 11$

Find the domain of $h(x) = \frac{1}{x - 4}$

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The domain is all real numbers except where the denominator equals zero.

Set the denominator equal to zero: $x - 4 = 0$ $x = 4$

We cannot divide by zero, so $x = 4$ must be excluded.

Answer: Domain: all real numbers except $x = 4$

In interval notation: $(-\infty, 4) \cup (4, \infty)$

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