Algebra
# 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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