Questions: Consider the function f(x) = 5^x and the transformation g(x) = 5^x - 2 (1) State the equation for the horizontal asymptote of g(x) (2) State the domain of g(x) using interval notation (3) State the range of g(x) using interval notation

Consider the function f(x) = 5^x and the transformation g(x) = 5^x - 2
(1) State the equation for the horizontal asymptote of g(x)
(2) State the domain of g(x) using interval notation
(3) State the range of g(x) using interval notation

Transcript text: Consider the function $f(x)=5^{x}$ and the transformation $g(x)=5^{x}-2$ (1) State the equation for the horizantal a symptote of $g(x)$ (2) State the domain of $g(x)$ using interval notation (3) State the range of $g(x)$ using interval notation

Solution

Solution Steps

To solve the given problems, we need to analyze the function $ g(x) = 5^x - 2 $.

Horizontal Asymptote: The horizontal asymptote of an exponential function $ f(x) = a^x + c $ is $ y = c $. For $ g(x) = 5^x - 2 $, the horizontal asymptote is $ y = -2 $.
Domain: The domain of an exponential function is all real numbers, since there are no restrictions on the values that $ x $ can take.
Range: The range of $ g(x) = 5^x - 2 $ is all real numbers greater than the horizontal asymptote. Since $ 5^x $ is always positive, $ 5^x - 2 $ will be greater than $-2$. Therefore, the range is $(-2, \infty)$.

Step 1: Determine the Horizontal Asymptote

The function $ g(x) = 5^x - 2 $ is an exponential function of the form $ a^x + c $. The horizontal asymptote for such a function is given by $ y = c $. Therefore, the horizontal asymptote of $ g(x) $ is:

\[ y = -2 \]

Step 2: Determine the Domain

The domain of an exponential function is all real numbers because there are no restrictions on the values that $ x $ can take. Thus, the domain of $ g(x) $ is:

\[ (-\infty, \infty) \]

Step 3: Determine the Range

The range of the function $ g(x) = 5^x - 2 $ is determined by the fact that $ 5^x $ is always positive. Therefore, $ 5^x - 2 $ will always be greater than $-2$. Hence, the range of $ g(x) $ is:

\[ (-2, \infty) \]