Question

The function f(x) = log₂ (-x+5) + 1 is graphed below. I already plotted the inverse points all I need is to figure out if it has a vertical or horizontal asymptote and the range and domain


So the answer is either (for the blank spaces) vertical or horizontal, x or y, whichever number matches with x and y and then the range and if it is decreasing or increasing and then the domain.

Answer 1

Answer:

It has a horizontal asymptote of y = 5.

The range of the function is (-∞, 5), and it is decreasing on its domain of (-∞, ∞).

Step-by-step explanation:

Original function

$f(x)=\log_2(-x+5) + 1$

For a logarithmic function to be defined, its argument must be greater than zero. Therefore:

$-x+5 > 0\\\\-x > -5\\\\x < 5$

So, the domain of f(x) is (-∞, 5).

The range of a logarithmic function is the set of all real numbers, so the range of f(x) is (-∞, ∞).

When x = 5, the argument of the given logarithmic function is equal to zero, which means there is a vertical asymptote at x = 5.

As x approaches 5 from the left, f(x) approaches negative infinity. As x approaches negative infinity, f(x) approaches positive infinity. Therefore, the function is decreasing on its domain of (-∞, 5).

$\hrulefill$

Inverse function

$f^{-1}(x)=-2^{(x-1)}+5$

The inverse of a function is the reflection of the original function across the line y = x, so essentially the roles of x and y are swapped. The domain of the original function corresponds to the range of the inverse function, and vice versa.

As the original function has a horizontal asymptote at x = 5, the inverse function has a vertical asymptote at y = 5.

The domain of the inverse function is (-∞, ∞), and its range is (-∞, 5).

As x approaches negative infinity, f(x) approaches 5. As x approaches positive infinity, f(x) approaches negative infinity. Therefore, the function is decreasing on its domain of (-∞, ∞).