SOLUTION: Suppose the domain of f is (-1,1). Define the function g by g(x) = 5

SOLUTION: Suppose the domain of f is (-1,1). Define the function g by g(x) = 5 - f(x) + f(5/x). What is the domain of g?

Question 1209612: Suppose the domain of f is (-1,1). Define the function g by
g(x) = 5 - f(x) + f(5/x).
What is the domain of g?

Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

f(5/x) is what we should focus on.
The input 5/x needs to be on the interval -1 < 5/x < 1, so it fits with the domain interval (-1,1).

You should find that -1 < 5/x < 1 leads to either x < -5 or x > 5.
But none of those values are in the interval -1 < x < 1.

Therefore the domain of g(x) is the empty set.

--------------------------------------------------------------------------

Some examples:

Let's try x = 0.5
g(x) = 5 - f(x) + f(5/x)
g(0.5) = 5 - f(0.5) + f(5/0.5)
g(0.5) = 5 - f(0.5) + f(10)
g(0.5) = 5 - f(0.5) + undefined
g(0.5) = undefined
This shows that x = 0.5 is not in the domain of g(x).
f(10) is undefined since f(x) is only defined when -1 < x < 1.

Let's try x = 10
g(x) = 5 - f(x) + f(5/x)
g(10) = 5 - f(10) + f(5/10)
g(10) = 5 - f(10) + f(0.5)
g(10) = 5 - undefined + f(0.5)
g(10) = undefined
We run into the same problem as before.
This shows that x = 10 is not in the domain of g(x).

I encourage the student to explore other examples.

RELATED QUESTIONS
Suppose the domain of f is (-1,3). Define the function g by g(x) = 5 - f(x) + f(5/x).... (answered by ikleyn)

Suppose the domain of f is (-1,3). Define the function g by g(x) = f((x + 1)(x - 2)).... (answered by CPhill)

Suppose the domain of f is (-1,3). Define the function g by g(x) = f(2/x + x/2). What... (answered by CPhill)

Suppose the domain of f is (-1,3). Define the function g by g(x) = f(sqrt(x)/sqrt(-x)). (answered by CPhill)

Suppose the domain of f is (-1,1). Define the function g by g(x) = f(x - 4x^2) What is... (answered by CPhill)

Suppose the domain of f is (-1,3). Define the function g by g(x) = f(x - 4x^2) What is... (answered by CPhill)

Suppose a function f(x) is defined on the domain [-8,4]. If we define a new function g(x) (answered by robertb)

========================================================================================== (answered by MathLover1)

3.21 Find f+g, f-g and f/g. Determine the domain for each function. f(x)=4x+8,... (answered by fractalier)