Question

heres an exponential function: ( y = -4^x )
(a) choose the graph that best represents the shape of this function.
(b) find the domain and range. write the answers as inequalities, using ( x ) or ( y ) as appropriate.
or, you may instead click on \empty set\ or \all reals\ as the answer.
domain:
range:
(c) give the ( x )-intercept(s) and ( y )-intercept(s). click on
one\ if applicable.
( x )-intercept(s):
( y )-intercept(s):
(d) the values of ( y ) are select for all values of ( x ).
(e) choose the best description for the end behavior.
( \bigcirc ) as ( x ) increases, ( y ) increases. as ( x ) decreases, ( y ) approaches ( 0 ).
( \bigcirc ) as ( x ) increases, ( y ) approaches ( 0 ). as ( x ) decreases, ( y ) increases.
( \bigcirc ) as ( x ) increases, ( y ) decreases. as ( x ) decreases, ( y ) approaches ( 0 ).
( \bigcirc ) as ( x ) increases, ( y ) approaches ( 0 ). as ( x ) decreases, ( y ) decreases.

Explanation:

Part (b) - Domain and Range

Step 1: Analyze the Domain of \( y = -4^x \)

The function \( y = -4^x \) is an exponential function. For any exponential function of the form \( a^x \) (where \( a>0, a
eq1 \)), the domain is all real numbers because we can substitute any real number for \( x \) and compute \( 4^x \) (and then multiply by -1). So the domain of \( y=-4^x \) is all real numbers, which in inequality form is \( x\in(-\infty, \infty) \) or written as \( x \) is all reals (we can use "All reals" as per the option).

Step 2: Analyze the Range of \( y = -4^x \)

First, recall the range of \( 4^x \). The function \( 4^x \) has a range of \( (0, \infty) \) because \( 4^x>0 \) for all real \( x \). Now, if we multiply \( 4^x \) by -1, we get \( -4^x \). Multiplying a positive number (since \( 4^x>0 \)) by -1 will give a negative number. So \( -4^x < 0 \) for all real \( x \). So the range of \( y = -4^x \) is \( y < 0 \) (or in interval notation \( (-\infty, 0) \)).

Step 1: Find the x-intercept

The x-intercept is the value of \( x \) when \( y = 0 \). So set \( y = 0 \) in the equation \( y=-4^x \):
\( 0=-4^x \)
But \( 4^x>0 \) for all real \( x \), so \( -4^x < 0 \) for all real \( x \). Therefore, there is no solution to \( 0=-4^x \), so the x-intercept is "None".

Step 2: Find the y-intercept

The y-intercept is the value of \( y \) when \( x = 0 \). Substitute \( x = 0 \) into \( y=-4^x \):
\( y=-4^0 \)
Since \( a^0 = 1 \) for \( a
eq0 \), \( 4^0 = 1 \). So \( y=-1 \). So the y-intercept is \( (0, -1) \), so the y-intercept (the value of \( y \) when \( x = 0 \)) is \( -1 \), so we can write the y-intercept as \( (0, -1) \) or just the y-value is -1 (but in the context of the problem, we can state the point or the value; since it asks for intercept(s), for y-intercept, when \( x = 0 \), \( y=-1 \), so the y-intercept is \( (0, -1) \), and x-intercept is "None".

Let's analyze the end behavior of \( y=-4^x \).

• As \( x \) increases ( \( x

ightarrow\infty \) ):
\( 4^x
ightarrow\infty \) (since the base 4 > 1, exponential growth), so \( -4^x
ightarrow -\infty \)? Wait, no, wait: Wait, \( 4^x \) as \( x
ightarrow\infty \) goes to \( \infty \), so \( -4^x \) goes to \( -\infty \)? Wait, no, wait, let's re-examine. Wait, no, the function is \( y = -4^x \). Let's take limits:

• As \( x

ightarrow \infty \): \( \lim_{x
ightarrow\infty} -4^x = -\infty \)? Wait, no, wait, \( 4^x \) grows to infinity, so \( -4^x \) goes to negative infinity? Wait, no, maybe I made a mistake earlier. Wait, no, let's check with values. Let \( x = 0 \), \( y=-1 \); \( x = 1 \), \( y=-4 \); \( x = 2 \), \( y=-16 \); \( x = 3 \), \( y=-64 \). So as \( x \) increases, \( y \) becomes more negative (decreases towards \( -\infty \)).

• As \( x

ightarrow -\infty \): \( \lim_{x
ightarrow-\infty} -4^x \). Let \( x = -n \) where \( n
ightarrow\infty \). Then \( 4^{-n}=\frac{1}{4^n}
ightarrow0 \) as \( n
ightarrow\infty \), so \( -4^{-n}
ightarrow0 \). So as \( x \) decreases (goes to \( -\infty \)), \( y \) approaches 0.

Now let's check the options:

1. As \( x \) increases, \( y \) increases. As \( x \) decreases, \( y \) approaches 0. → No, \( y \) decreases as \( x \) increases.
2. As \( x \) increases, \( y \) approaches 0. As \( x \) decreases, \( y \) increases. → No.
3. As \( x \) increases, \( y \) decreases. As \( x \) decreases, \( y \) approaches 0. → Yes, this matches.
4. As \( x \) increases, \( y \) approaches 0. As \( x \) decreases, \( y \) decreases. → No.

Answer:

(for part b):
Domain: All reals
Range: \( y < 0 \)

Part (c) - x-intercept and y-intercept