Question

If `f(2x+1)=2x-1` for all real numbers x, then `f(x)` = ?

i have no idea how to do this, please help!

Answer 1

`f(x) = x - 2`

Explanation:

the difference between `2x+1` and `2x-1` is `2`.

`(2x-1) - (2x + 1) = 2x-2x-1-1 = -2`

`(2x-1) - (2x + 1) = -2`

when `2x+1` is the input of the equation, `2x - 1` is the output.

this means that when `x` is the input, then `x - 2` is the output.

this can also be seen on a graph:

![desmos.com/calculator]()

the graph in red is the graph of `f(x)`, where `f(x) = x - 2`

the graph in blue is the graph of `f(2x+1)`, where `f(x) = x-2`.

if you look at the blue graph along the `x`-axis from `1` to `2`, then you'll see that the graph goes across by `1` unit, `(1 rarr 2)` and up by `2` units `(1 rarr 3)`.

the gradient of the blue graph is therefore `2/1`, which is `2`.

it can also be seen that the blue line crosses the `y`-axis when `y` is at `-1`. this means that the `y`-intercept of the graph is `-1`.

all linear graphs can be written in `y = mx + c` form, where `m` is the gradient of the graph and `c` is the `y`-intercept.

here, `m = 2` and `c = -1`, so the equation of the graph in blue is `y = 2x-1`.

the blue line is the graph of `f(2x+1)`, so here, the equation of `f(2x + 1) = 2x - 1`.

this is only true when `f(x) = x - 2`, which is the graph shown in red.