QuestionUse the function $f(x)=3^{x-1}-5$ to answer the following: (a) Determine the equation of the asymptote of $f(x)$ .
Your answer should include $y=$ or $x=$ $\square$ (b) Determine the domain of $f(x)$ in interval notation. $\square$ (c) Determine the range of $f(x)$ in interval notation. $\square$

Studdy Solution

STEP 1

1. The function $f(x) = 3^{x-1} - 5$ is an exponential function.
2. Exponential functions of the form $a^{x}$ have horizontal asymptotes.
3. The domain of exponential functions is typically all real numbers.
4. The range of an exponential function $a^{x} + c$ is affected by the vertical shift $c$ .

STEP 2

1. Determine the equation of the asymptote.
2. Determine the domain of $f(x)$ .
3. Determine the range of $f(x)$ .

STEP 3

To find the horizontal asymptote of the function $f(x) = 3^{x-1} - 5$ , we consider the behavior of the exponential part $3^{x-1}$ as $x$ approaches negative infinity.
For exponential functions of the form $a^{x}$ , as $x \to -\infty$ , $a^{x} \to 0$ .
Thus, the horizontal asymptote is determined by the constant term in the function, which is:
$y = -5$

STEP 4

The domain of the function $f(x) = 3^{x-1} - 5$ is determined by the values of $x$ for which the function is defined.
Since exponential functions are defined for all real numbers, the domain is:
$(-\infty, \infty)$

STEP 5

To determine the range of $f(x) = 3^{x-1} - 5$ , we consider the transformation of the basic exponential function $3^{x}$ .
The function $3^{x-1}$ is a horizontal shift of $3^{x}$ , and subtracting 5 shifts the entire graph down by 5 units.
The range of $3^{x}$ is $(0, \infty)$ , so the range of $3^{x-1} - 5$ is:
$(-5, \infty)$
(a) The equation of the asymptote is $y = -5$ . (b) The domain of $f(x)$ is $(-\infty, \infty)$ . (c) The range of $f(x)$ is $(-5, \infty)$ .

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