Questions: Use transformations to graph the function. Determine the domain, range, horizontal asymptote, and y-intercept of the function. f(x)=4+5^(x-1)

Solution Steps

The function \( f(x) = 4 + 5^{x-1} \) is defined for all real numbers \( x \). Therefore, the domain is: \[ \text{Domain: } (-\infty, \infty) \]

The expression \( 5^{x-1} \) is always positive for any real \( x \), and as \( x \to -\infty \), \( 5^{x-1} \to 0 \). Therefore, the range of \( f(x) = 4 + 5^{x-1} \) is: \[ \text{Range: } (4, \infty) \]

As \( x \to -\infty \), \( 5^{x-1} \to 0 \), so \( f(x) \to 4 \). Thus, the horizontal asymptote is: \[ y = 4 \]

To find the \( y \)-intercept, set \( x = 0 \): \[ f(0) = 4 + 5^{0-1} = 4 + \frac{1}{5} = 4.2 \] Thus, the \( y \)-intercept is: \[ (0, 4.2) \]

Final Answer