help... 3 questions

Question

1) Find the inverse of the function y = - 4. *show your work*

2) Determine algebraically of y = x^2 + 8 is a one-to-one function. *show your work*

3) Rewrite 6^2 = 36 as a logarithmic function. *show your work*

Answer 1

1) y = -4 does not have an inverse function. If, say, x=1, y=-4. If x=2, y=-4. No matter what the value of x, y is always -4, so, given an output (in this case, every output is -4), there is not a unique corresponding input.

For a visual perspective, the graph of y=-4 is a horizontal line, which does not pass the "horizontal line test"

2) There are different ways to think about this. 1-1 means that different inputs always give different outputs, or, equivalently, a given output only has one input that will yield it. Here's one approach:

Assume that y is given. Is there a unique x that would give us that y? Well, use the equation to try and solve for x:

y = x^2+8, so

y-8 = x^2, hence

x = ±sqrt(y-8).

This tells us that, as long as y ≥8, there are two x values that yield the same y value, hence y is not a one-to-one function.

3) y = b^x is equivalent to log_b(y) = x

So, 'translating' the given expression, 6^2 = 36 is equivalent to log_6(36) = 2