Question

piecewise function f (x) is defined by f of x is equal to the piecewise function of 3 to the power of the quantity x minus 1 end quantity minus 4 for x is less than or equal to 3 and the quantity negative x squared plus 3 times x plus 4 end quantity over the quantity x squared minus 7 times x plus 12 end quantity for x is greater than 3 Part A: Graph the piecewise function f (x) and determine the range. (5 points) Part B: Determine the asymptotes of f (x). Show all necessary calculations. (5 points) Part C: Describe the end behavior of f (x). (5 points)

Answer 1

The range of the function f(x) is (-∝, 3]

Part A: Graph the piecewise function

The function definition is given as:

$f(x) = \left[\begin{array}{cc}3^{x-1}-4&x\le 3\\ \frac{-x^2 + 3x + 4}{x^2 - 7x + 12}&x > 3\end{array}\right$

There are two sub-functions and the domains in the above definition.

Each function would be plotted alongside its domain.

See attachment for the graph of the function f(x)

From the graph of the function, we have the following range of f(x)

Minimum = Negative Infinity

Maximum = 5

Hence, the range of the function f(x) is (-∝, 3]

The asymptotes of f(x)

We have the domains to be

x <= 3 and x > 3

This means that the asymptote of f(x) is x = 3

The end behavior of f(x)

From the graph, we have:

• f(x) increases as x increases
• f(x) decreases as x decreases

This means that the end behavior of f(x) is as x approaches +∝, the function approaches +∝ and as x approaches -∝, the function approaches -∝

Read more about functions at:

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Complete question

A piecewise function f (x) is defined by

$f(x) = \left[\begin{array}{cc}3^{x-1}-4&x\le 3\\ \frac{-x^2 + 3x + 4}{x^2 - 7x + 12}&x > 3\end{array}\right$