Questions: Functions Evaluating functions: Absolute value, rational, radical The functions f, g, and h are defined as follows. f(x) = x / (4 + x^2) g(x) = sqrt(3x + 9) h(x) = -12 + 9x Find f(-4), g(2), and h(-1/3). Simplify your answers as much as possible. f(-4) = g(2) = h(-1/3) =

Functions
Evaluating functions: Absolute value, rational, radical

The functions f, g, and h are defined as follows.

f(x) = x / (4 + x^2) g(x) = sqrt(3x + 9) h(x) = -12 + 9x

Find f(-4), g(2), and h(-1/3).
Simplify your answers as much as possible.

f(-4) =
g(2) =
h(-1/3) =

Transcript text: Functions Evaluating functions: Absolute value, rational, radical The functions $f, g$, and $h$ are defined as follows. \[ f(x)=\frac{x}{4+x^{2}} \quad g(x)=\sqrt{3 x+9} \quad h(x)=-12+|9 x| \] Find $f(-4), g(2)$, and $h\left(-\frac{1}{3}\right)$. Simplify your answers as much as possible. \[ \begin{array}{r} f(-4)= \\ g(2)= \\ h\left(-\frac{1}{3}\right)= \end{array} \]

Solution

Solution Steps

Step 1: Calculate $ f(-4) $

The function $ f(x) $ is defined as: \[ f(x) = \frac{x}{4 + x^2} \] Substitute $ x = -4 $ into the function: \[ f(-4) = \frac{-4}{4 + (-4)^2} = \frac{-4}{4 + 16} = \frac{-4}{20} = -\frac{1}{5} \]

Step 2: Calculate $ g(2) $

The function $ g(x) $ is defined as: \[ g(x) = \sqrt{3x + 9} \] Substitute $ x = 2 $ into the function: \[ g(2) = \sqrt{3(2) + 9} = \sqrt{6 + 9} = \sqrt{15} \]

Step 3: Calculate $ h\left(-\frac{1}{3}\right) $

The function $ h(x) $ is defined as: \[ h(x) = -12 + |9x| \] Substitute $ x = -\frac{1}{3} $ into the function: \[ h\left(-\frac{1}{3}\right) = -12 + \left|9 \left(-\frac{1}{3}\right)\right| = -12 + | -3 | = -12 + 3 = -9 \]