Question

1. label the graphs as f(x) or g(x).

f(x) = -1.5^x and g(x) = -2.7^x
graph
a:
______
b:
______
a. f(x)
b. g(x)

1. solve. round each calculation to the nearest hundredth. you will need a scientific or graphing calculator.

francium has a decay rate of 50% every 22 minutes. how much of an initial amount of 4.8 grams would be le
2.4 grams
4.8 grams

Explanation:

Step1: Analyze the base of the exponential functions

For \( f(x) = -1.5^x \) and \( g(x) = -2.7^x \), the bases are \( 1.5 \) and \( 2.7 \) respectively, and \( 1.5<2.7 \).

Step2: Analyze the behavior of the functions for \( x>0 \)

For exponential functions of the form \( y = -a^x \) (\( a>1 \)), as \( x \) increases, the function values become more negative (decrease) faster for larger \( a \). But when \( x>0 \), let's take a value of \( x \), say \( x = 2 \).
For \( f(2)=-1.5^2=-2.25 \)
For \( g(2)=-2.7^2=-7.29 \)
Wait, but in the graph, we can also look at the steepness. Wait, actually, when \( x \) is positive, the function with the smaller base will be "less negative" (closer to zero) than the one with the larger base. So for \( x>0 \), \( f(x) \) (with base 1.5) is greater than \( g(x) \) (with base 2.7) because \( -1.5^x>-2.7^x \) when \( x>0 \) (since \( 1.5^x < 2.7^x \), multiplying by -1 reverses the inequality).
Looking at the graph, the curve labeled A is less steep (closer to the x - axis for positive x) than curve B. So A should correspond to the function with the smaller base's negative exponential, which is \( f(x)=-1.5^x \), and B corresponds to \( g(x)=-2.7^x \). Wait, no, wait: when \( x = 0 \), both \( f(0)=-1.5^0=-1 \) and \( g(0)=-2.7^0=-1 \), so they intersect at \( (0, - 1) \). For \( x = 1 \), \( f(1)=-1.5 \), \( g(1)=-2.7 \). So \( f(1)>-g(1) \)? Wait, \( -1.5>-2.7 \), so at \( x = 1 \), \( f(x) \) is greater (closer to zero) than \( g(x) \). So the graph that is higher (closer to zero) for \( x>0 \) is \( f(x) \), and the lower one is \( g(x) \). In the given graph, curve A is higher (closer to zero) than curve B for \( x>0 \), so A is \( f(x) \) and B is \( g(x) \).

Step3: Confirm with the options

Option a is \( f(x) \) and option b is \( g(x) \). So A should be \( f(x) \) (a) and B should be \( g(x) \) (b). Wait, no, wait the options are a. \( f(x) \), b. \( g(x) \). So A: \( f(x) \) (a), B: \( g(x) \) (b). Wait, let's re - check.
When \( x = 1 \), \( f(1)=-1.5 \), \( g(1)=-2.7 \). So \( f(1)>-2.7 \), so the graph of \( f(x) \) is above (closer to zero) the graph of \( g(x) \) at \( x = 1 \). In the given graph, the curve labeled A is above curve B at \( x = 2 \) (looking at the graph, A is less steeply decreasing), so A is \( f(x) \) and B is \( g(x) \).

Answer:

A: a. \( f(x) \)
B: b. \( g(x) \)