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Find the slope of a line parallel to 4x+2y=5

Aarav Atkins 2022-10-29

Find all functions f such that $f (x - f (y)) = f (f (x)) - f (y) - 1$
So far, I've managed to prove that if f is linear, then either $f (x) = x + 1$ or $f (x) = - 1$ must be true. I did this by plugging in $x = 0$ to the above equation, which yields
$f (- f (y)) = f (f (0)) - f (y) - 1$
and plugging in x instead of y and subtracting, this becomes
$f (- f (y)) - f (- f (x)) = f (x) - f (y)$
Assuming $f (x) = a x + b$ then gives
$\begin{aligned} f (- a y - b) - f (- a x - b) = a x - a y & \Rightarrow - a^{2} y - a b + a^{2} x + a b = a (x - y) \\ \Rightarrow a^{2} (x - y) = a (x - y) . \end{aligned}$
Thus $a = 1$ or $a = 0$ . If $a = 0$ , then the original equation becomes $b = b - b - 1$ , thus $b = - 1$ . If $a = 1$ , the original equation becomes
$x - y - b + b = x + 2 b - y - b - 1 ⟹ b = 1.$
I briefly tried finding a quadratic function that works but didn't find anything. So my question is: how can I either show that f must be linear or find all other representations?

Kasey Reese 2022-10-29

Write the equation in point slope form given (3,5) and m=-2

djo57bgfrqn 2022-10-29

What is the 5th term of the sequence if the first term of a geometric sequence is 320, and the common ratio is 0.5

Alessandra Cummings 2022-10-29

Write 0=x+5 in slope intercept form

Iris Vaughn 2022-10-29

8 people meet, and each person shakes every other person's hand once. How many handshakes occurblack?

klastiesym 2022-10-29

Can we say $f (x) = \frac{x}{\ln (x^{2})}$ has a removable discontinuity at $x = 0$ ?

4enevi 2022-10-29

Find nth term rule for $a_{2} = 28$ and $a_{5} = - 1792$

pezgirl79u 2022-10-29

Find the slope that is perpendicular to the line y=−4x+2

Nayeli Osborne 2022-10-29

Find the 10th term in the geometric sequence 320, 160, 80, 40, ...

Maverick Avery 2022-10-29

Find the nth term of the geometric sequence 5,15,45,135

omgespit9q 2022-10-29

Write an equation of a line given point (4, -1) and perpendicular to a line with equation 3x + y =4

Hugo Stokes 2022-10-29

Write an equation of a line parallel to y=−3x+4 and x intercept at 4

Oscar Burton 2022-10-29

Write an equation of a line given (12,3) and perpendicular to 3x-y=2

Valery Cook 2022-10-29

Identify the 15th term of the geometric sequence 52, 39, 29.25 …

erwachsenc6 2022-10-29

A one-ton ice sculpture is melting so that is loses one-fifth of its weight per hour. How much of the sculpture will be left after five hours?

beefypy 2022-10-29

Write an equation of a line passing through (0, 4), perpendicular to y=6x−2

Mariyah Bell 2022-10-29

if we take an irrational number like $π$ or $e$ or whatever and we write this $π + π - π \dots$ (ecc) at infinity of this series what could possibly come out?

Tessa Peters 2022-10-28

Find the indicated term for the geometric sequence $a_{1} = 7$ , r=2, n=4

klastiesym 2022-10-28

Find the slope of the line parallel to y=4x+2

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