Question

Consider f(x) = bx. Which statement(s) are true for 0 < b < 1? Check all that apply:

The domain is all real numbers.

The domain is x > 0.

The range is all real numbers.

The range is y > 0.

The graph has x-intercept 1.

The graph has a y-intercept of 1.

The function is always increasing.

The function is always decreasing.

Answer 1

we are given

$f(x)=b^x$

where 0<b<1

Domain:

domain is all possible values of x for which any function is defined

so, we can select any values of x for which function

it will be defined for all real x

so, domain is

$(-\infty,\infty)$

Range:

Range is all possible values of y for which x is defined

we are given that b is positive

so, value of function will always be positive

so, range is

$(0,\infty)$

or

y>0

x-intercept:

we can set f(x)=0

and then we can solve for x

$f(x)=b^x=0$

x is undefined

so, x-intercept does not exist

Increasing or decreasing:

Since, 0<b<1

so, b is positive value less than 1

so, as we increase value of , b^x will keep decreasing

so, this is decreasing

Answer 2

The true statements about the function are:

• (a): The domain is all real numbers.
• (d): The range is y > 0.
• (h): The function is always decreasing.

The function is given as:

$\mathbf{f(x) = b^x}$

$\mathbf{0 < b < 1}$

There is no restriction on the values of x.

So, the domain of the function is: (a) the set of all real numbers

$\mathbf{0 < b < 1}$ means that, the smallest value of the function will be greater than  0, and the function will not cross the x-axis

So, the range of the function is: (d) y > 0, and the graph has no x-intercept

$\mathbf{0 < b < 1}$ also means that the value of b is a positive number less than 1.

So, as x increases, f(x) decreases

In other words, (h) the function is always decreasing.

Hence, the true statements about the function are: (a), (d) and (h)

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