F(x) = b^x and g(x) = log b x are inverse functions. explain why each of the following are true.

1. a translation of function f is f1(x) = b^(x-h). it is equivalent to a vertical stretch or vertical compression of function f.

2. the inverse of f1(x) = b^(x-h) is not equivalent to a translation of g.

3. the inverse of f1 (x) =b^(x-h) is not equivalent to a vertical stretch or vertical compression of g.

4. the function h(x) = log c x is a vertical stretch or compression of g or of its reflection -g. read this as"negative g"

will probably needs to use the properties of exponents and logarithms and change of base formulas to change the functions into alternate forms

1. A translation of function f is . It is equivalent to a vertical stretch or vertical compression of function f.

Let's take the function:

If we used the rules of powers, we know that when we have a subtraction in the power of an exponential, it can be split into a division. So the function can be rewritten as:

Remember the original function was:

therefore:

this means that if then it will be a vertical stretch.

If then it will be a vertical stretch.

2. The inverse of is not equivalent to a translation of g.

This is partially true and you'll see why. Let's start by finding the inverse of that function:

we start by turning the given power to a multiplication of powers so we get:

we then move the to the other side of the equation so we get:

and turn the equation into a logarithm:

so:

or:

remember that when you multiply a constant by x, you will get a horizontal compression if and a horizontal stretch if .

but there is another interpretation for this function. Let's take the original equation:

if we directly turned this equation into a logarithm we would get that:

so:

or:

if the inverse function is written like this, it can be interpreted as a vertical shift. Both interpretations are correct.

3. The inverse of is not equivalent to a vertical stretch or vertical compression of g.

As we saw in the previous part of the problem, that function is either a horizontal stretch/compression or a vertical shift, not a vertical stretch or compression.

4. The function is a vertical stretch or compression of g or of its reflection -g.

We can rewrite the function like this thanks to log rules:

which is the same as:

If it will be a vertical compression of g(x). If , then it will be a vertical stretch no matter if g is positive or negative.

what's that