Alan

Sorry, I misread the second part of your question.  However, 2^-x is not very different from your original f(x), so if you shift the red graph up by 1 everywhere you should find the intersection.

Better than this is to note that 2^-x = f(x)+1, so if 2^-x = x + 3 then f(x) + 1 = x + 3 or f(x) = x + 2.  Lower the straight (blue) line by 1 everywhere (easier than increasing the red line by 1) to find the intersection point. Looks like the two curves would then intersect at x = -1.

Check:  LHS = 2^(- -1) = 2^(+1) = 2.  and RHS = -1+3 = 2

So this checks out ok!

My earlier comment about not being able to isolate x still stands.

In terms of quadrants, the graph of b^x - 1 lies in the 2nd (x negative, y positive) and 4th (x positive, y negative) quadrants.  The graph of x + 3 lies in the 1st (x positive, y positive), 2nd and 3rd (x negative, y negative) quadrants.  The intersection of these two functions lies in the 2nd quadrant.

Your original question specified using the graph (of the intersection of the two functions) to find the value of x at the point of intersection.  Since you originally spoke of sketching the curve, I assumed you would not be after a very precise value (you can see from the graph that the intersection occurs close to x = -1.4).

When equating the two functions you can't rearrange the result to isolate x.  You need to solve the resulting equality numerically (to get -1.386...).  (Technically, there is an analytical solution, but it involves a non-elementary function called the LambertW function - not something most of us meet in every day life!)

That's the right way!

This might help  (note that (1/2)^x is the same thing as 2^-x):

.

heureka has given the correct answer to the question that was asked.

If you think it is wrong you need to say why.

An infinite number of lines go through (3, -1).  You need to know the slope of the straight line as well before you can find the intercept.  

See below:

y = 5^x

y^2 = (5^x)^2 = 5^(2x)

5y^2 = 5*5^(2x) = (5^1)*(5^2x) = 5^(2x+1)

As long as b) is meant to be 5^(2x+1) not 5^(2x) + 1  then b) is the answer.  Please use brackets to ensure an unambiguous interpretation.

3^m+3^m+3^m → 3*3^m → (3^1)*(3^m) → 3^(m+1)

If we assume there are 8 pieces of paper with an 'a' on them, 8 pieces with a 'b', 8 pieces with a 'c' and 7 pieces with a 'd', 7 with an 'e', 7 with an 'f' and 7 with a 'g', then there are 3*8 + 4*7 = 52 pieces of paper in total, 8 of which have a 'c'.  The probability of drawing a piece with a 'c' on it is therefore 8/52 = 2/13 ≈ 0.154