Question

Show that the function f(x) is a growth function for x<5 .

f(x)=5-x/5^-x

Answer 1

\(\color{#000000}{ \displaystyle f(x)=\frac{5-x}{5^{-x}} }\) this?

Answer 2

yes sir

Answer 3

OK. I would assume that "growth function for t>5" means that the function is increasing for (all) t>5.

Answer 4

We can actually legitimately prove this using calculus, if this is a calculus question. Is it calculus?

Answer 5

no sir algebra 2

Answer 6

Alright.

Answer 7

Well, then we can't "prove" the hypothesis.
However, you could show that the function is going to be positive, and the amount will be always increasing for all x>5.

Answer 8

I will first re-write the function as,

\(\color{#000000}{ \displaystyle f(x)=(5-x)5^{x} }\)

Answer 9

Do you understand how I was able to write the f(x) this way?

Answer 10

no sir

Answer 11

There is a rule:
\(\color{#000000}{ \displaystyle a^{-b}=\frac{1}{a^b} }\)

So, here I would get:
\(\color{#000000}{ \displaystyle f(x)=\frac{5-x}{5^{-x}} }\)

\(\color{#000000}{ \displaystyle f(x)=(5-x)\times \frac{1}{5^{-x}} }\)

\(\color{#000000}{ \displaystyle f(x)=(5-x)\times \frac{1}{\frac{1}{5^x}} }\)

\(\color{#000000}{ \displaystyle f(x)=(5-x)\times 5^x }\)

Answer 12

How about now?

Answer 13

yes sir i understand

Answer 14

Ok, so your function, really is:
\(\color{#000000}{ \displaystyle f(x)=(5-x)5^x }\)

And you are asked to show that when x <5, the function is increasing.

Answer 15

all it says is to show that the fuction f(x) is a growth function for x<5

Answer 16

Yes, and I am interpreting that wording, as showing that the function is increasing.

Answer 17

oh ok continue sir

Answer 18

\(\color{#000000}{ \displaystyle f(x)=(5-x)5^x }\)

I can do this by plugging in some values.
(This is not really a proof, because the real proof is calculus, however it is a way of showing the hypothesis is true, so let's begin ...)


\(\color{#000000}{ \displaystyle f(\color{red}{-3})=(5-(\color{red}{-3}))5^{\color{red}{-3}} =8\cdot \frac{1}{5^3}=\frac{8}{125} }\)

\(\color{#000000}{ \displaystyle f(\color{red}{-2})=(5-(\color{red}{-2}))5^{\color{red}{-2}} =7\cdot \frac{1}{5^2}=\frac{7}{25} }\)

and so on ...

Answer 19

Plug 4 or 5 values into f(x) (that are smaller than 5), and you will see that the smaller the input the smaller the output (and the bigger the input, the bigger the output.)

*** (((Note that for example, -6 is less than -3, and -2 is less than 1 ....
and that should precisely clarify what I mean by "bigger" and "smaller")))

Answer 20

when you get --> the bigger the input the bigger the output,
you may than conclude that the function (for x<5) increases
(or, "a growth function")

Answer 21

Also, if you want a similar example, let me know.

Answer 22

i have another question i will post it will you help me btw god bless you and thanks

Answer 23

Not a problem.
If I have time, I'll take a look:)