Question

Find \((f \circ g)(x)\) and \((g \circ f)(x),\) and graph each of these functions. $$ \begin{array}{l} f(x)=\tan x \\ g(x)=4 x \end{array} $$

Short answer

(f ∘ g)(x) = tan(4x), (g ∘ f)(x) = 4tan(x)

Step-by-step solution

Define the Given Functions

The given functions are \[ f(x) = \tan(x) \] and \[ g(x) = 4x \].

Compute (f ∘ g)(x)

To find \((f \, \circ \, g)(x)\), you substitute \(g(x) = 4x\) into \(f(x) = \tan(x)\). \[(f \, \circ \, g)(x) = f(g(x)) = f(4x) = \tan(4x)\]

Compute (g ∘ f)(x)

To find \((g \, \circ \, f)(x)\), you substitute \(f(x) = \tan(x)\) into \(g(x) = 4x\). \[(g \, \circ \, f)(x) = g(f(x)) = g(\tan(x)) = 4 \, \tan(x)\]

Graph (f ∘ g)(x)

The function \((f \, \circ \, g)(x) = \tan(4x)\) can be graphed. The graph will look like the tangent function but with a period \(\pi/4\) instead of \(\pi\).

Graph (g ∘ f)(x)

The function \((g \, \circ \, f)(x) = 4 \, \tan(x)\) can be graphed. The graph will look like the tangent function but with a vertical stretch by a factor of 4.

Key concepts

Composite Functions

Composite functions combine two functions into one by plugging one function into the other. In this exercise, we looked at two given functions: \[ f(x) = \tan(x) \] and \[ g(x) = 4x \] We needed to find both \[ (f \circ g)(x) \] and \[ (g \circ f)(x). \] Here's how:

• To find \[ (f \circ g)(x), \] you substitute \[ g(x) \] into \[ f(x). \] So you get \[ f(g(x)) = f(4x) = \tan(4x). \]
• To find \[ (g \circ f)(x), \] you substitute \[ f(x) \] into \[ g(x). \] This gives you \[ g(f(x)) = g(\tan(x)) = 4 \tan(x). \]

These new functions show how composing functions works by chaining their operations.

Tangent Function

The tangent function, written as \[ \tan(x), \] is one of the main trigonometric functions. It represents the ratio of the opposite side to the adjacent side of a right triangle in a unit circle. The standard tangent function has:

• A period of \[ \pi, \] meaning the function repeats every \[ \pi \] units.
• Vertical asymptotes at \[ x = \frac{\pi}{2} + n\pi, \] where \[ n \] is an integer.

For \[ (f \circ g)(x) = \tan(4x), \]
The period changes to \[ \frac{\pi}{4} \] because the function inside \[ \tan \] is scaled by 4. This causes the tangent function to repeat more frequently. Each segment of the graph spans one-fourth of its original period.

Graph Transformations

Transforming the graph of a function involves shifting, stretching, compressing, or flipping it. For the given functions:
\[ (f \circ g)(x) = \tan(4x) \]

• Compresses the regular tangent graph horizontally, reducing the period to \[ \pi/4. \] The vertical asymptotes now occur at \[ x = \frac{\pi}{8} + n \frac{\pi}{4}. \]

\[ (g \circ f)(x) = 4 \tan(x) \]

• Stretches the tangent graph vertically by a factor of 4, making the peaks and troughs four times higher and lower than the original.

Graph transformations are powerful tools for understanding changes in function behaviors due to alterations in their formulas.