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# Functions

Functions map an input to an output. For example, the function f(x) = 2x takes an input, x, and multiplies it by two. An input of x = 2 gives you an output of 4. Learn all about functions.

# Evaluating Functions

If $$f(x) = 2x,$$ what is the value of $$f(-4)?$$

If $$f(x) = ax,$$ what is the value of $$a$$ satisfying $$f(-1) = 3?$$

Two functions $$f$$ and $$g$$ satisfy the following relationships.

\begin{align} f(1)= 2 \\ f(2)= 1 \\ f(3)= 2 \\ f(4)= 5 \\ g(1)= 3 \\ g(2)= 5 \\ g(3)= 7 \\ g(4)= 9 \\ \end{align}

What is the value of $$(f \circ g)(1)$$ ?

Let $$\mathbb{R}^+$$ be the set of all positive real numbers, and let $$f:\mathbb{R}^+ \to \mathbb{R}$$ be defined by

$f(x)= \begin{cases} \frac{1}{q} & \text{ if } x= \frac{p}{q}, \\ 0 & \text{ if } x \text{ is an irrational number}, \end{cases}$

where $$p$$ and $$q$$ are coprime positive integers. What is the value of $f\left(\frac{12}{13}\right)+f(\pi)+f(0.75)?$

If $$f(x) = 2x + a,$$ what is the value of $$a$$ satisfying $$f(2) = 2?$$

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