Question

What is the range of the function y=e^4x?

y<0
y>0
y<4
y>4

Answer 1

We have been given the function $y=e^{4x}$

We know that the range is set of y values for which the function is defined. Therefore, we will find the value for x and then observe the restriction is y's values.

$y=e^{4x}\\\ln y = \ln (e^{4x})\\\ln y =4x\\x=\frac{1}{4}\ln y$

Now, we know that logarithm function is not defined for negative values. Hence, the value for y is always greater than zero.

Therefore, the range of the function is given by y>0

B is the correct option.

Answer 2

You can use the fact that negative power can be written as that number raised with positive power but now in denominator with numerator being 1.

The range of the given function is

Option B: y > 0

What is domain and range of a function?

• Domain is the set of values for which the given function is defined.
• Range is the set of all values which the given function can output

How can we rewrite a number raised to negative power?

Suppose we've got $a^{-b}$

Then it can be rewritten as

$a^{-b} = \dfrac{1}{a^b}$

It is because

$a^0 = 1$

for any $a$ except 0, and that

Using the above conclusions to get the range of the given function:

Since exponential functions are defined on all real numbers(its domain is all real numbers), and as power increases, the value of function increases and can reach to +infinity, thus, the range's maximum value is +infinity. The function given is continuous.

We've to find minimum value.

For that, we will put x < 0 so that the right hand side of the given function under 1.

We get:

$x < 0 \implies y = \dfrac{1}{e^{4 \times |x|}}$

where |x| means getting only positive value of x as use of sign was done already to make it go under 1.

Now, as x goes more and more negative, the denominator increases more and more and the value of y will tend to 0. But it won't become 0 ever (it will tend to zero. Tending towards something is different than being that thing).

Also, since there is no chance of anything going negative, thus, we have output y always staying above 0(y can be infinitesimally close to 0 but not exact 0).

Thus,

The range of the given function is

Option B: y > 0

Learn more about domain and range of functions here:

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