Respuesta :
Given equation is [tex] f(x)=5 (6)^x [/tex]
Now we need to choose correct statement(s) from the given choices.
(A): It is always decreasing.
Given function is an exponential function with growth factor (6) which is more than 1. Hence it will be increasing function.
So this choice is FALSE.
(B): The y-intercept is (0, 5).
y-intercept can be found by plugging x=0 into given equation
[tex] f(x)=5(6)^0=5(1)=5 [/tex]
which same as given y-intercept (0,5)
So this choice is TRUE.
(C): The y-intercept is (0, 1).
As shown above, y-intercept is (0,5) not (0,1)
So this choice is FALSE.
(D): The domain of f(x) is x > 0
For any exponential function, domain is all real number not just x>0,
So this choice is FALSE.
Answer:
Option B - The y-intercept is (0, 5).
Step-by-step explanation:
Given : The function [tex]f(x)=5\cdot 6^x[/tex]
To find : Which of these statements is true for function ?
Solution :
The exponential function is in the form [tex]f(x)=ab^x[/tex]
where, a is the initial amount and b is the growth/decay rate factor.
and [tex]a\neq 0,\ b>0,\ b\neq 1[/tex] and x is any real number.
To check the statement is true or not,
A) It is always decreasing.
Function [tex]f(x)=5\cdot 6^x[/tex]
An exponential function is increasing when b>1,
As 6>1 it is increasing function.
So, It is false.
B) The y-intercept is (0, 5).
y-intercept means the value of x is 0.
[tex]f(0)=5\cdot 6^0[/tex]
[tex]f(0)=5[/tex]
The y-intercept is (0,5).
So, It is true.
C) The y-intercept is (0, 1).
The y-intercept is (0,5) solved in part B.
So, It is false.
D) The domain of f(x) is x > 0.
Domain is defined as the possible value of x where function is defined.
The domain of exponential functions is all real numbers.
So, It is false.
Therefore, Option B is correct.
