Question

Can y’all do 1-2 I’ll give brainliest !!!

Answer 1

One to one or injective means that each x in the domain maps to a unique y, i.e. the function preserves distinction.

y=(x-1)/(3x+3) is one to one.  We can solve for x:

3xy + 3y = x - 1

3y + 1 = x - 3xy = x(1-3y)

x = (3y+1)/(1-3y)

That proves it's one-to-one; for each y we get one x.

$y=\sqrt{5x+9}$

That's also one to one,

$y^2 = 5x+9$

$x = \frac 1 5 (y^2 - 9)$

That's the proof.

$f(x)=\dfrac{7}{4x^2}$

That's Not one-to-one

because f(-x)=f(x).  We only need a single counterexample; f(1)=7/4, f(-1)=7/4

$f(x) = \frac 1 2 x^3$

That's one-to-one, textbook case

$f(x)=3x^4 + 7x^3$

That's not one-to-one because the fourth power dominates so this will be sort of parabola-ish, repeating the same big positive values for a negative and positive x.

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$y = f(x) = \sqrt[3]{x-2} + 8$

$y - 8 = \sqrt[3]{x-2}$

$(y-8)^3 = x-2$

$x = (y - 8)^3 + 2$

$f^{-1}(x) =(x - 8)^3 + 2$

First choice