MAIN CONCEPTS
First be careful to apply the function correctly, bearing in mind we are substituting the function in the x on the right hand side.
From , we solve c by eliminating x.
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This question gets a little tricky because at first sight, it seems that we are solving for two variables, x and c, with only one equation. In a way, it seems that way, but the truth is that the variable we want to solve is c as posed by the question. What I suspect would happen is that we will find an equation to solve and somehow ‘eliminate’ x from that equation. That equation will come from the given .

Here is where my tip with dealing with functions comes in handy. It is vital that you substitute the ‘function of a function’ properly. My tip is to rewrite the function for a clearer picture.

Basically, we will put whatever is in the blank in the ‘( )’ into the blanks on the right hand side. So, it should be clear that

Given that this is also equal to , we have the equation

 

or

This is where our anticipated problem occurs, how are we going to solve for both c and x from this one equation. Well, with some algebraic manipulation, we can get through this problem.

The equal sign plays an important role here. If we can somehow pick a value of c such that we can turn both the coefficients of  and  to zero, we would have made the left hand side equal to the right hand side, regardless the value of x. Let’s try. We set  giving us  and  giving us . Thus, we know that  as it satisfies the equation.

So our final answer is . I shall comment a little in saying that this works because we are at ‘liberty’ to pick a value for c which leads to us setting the appropriate equations.

Now, here comes an intriguing part. Another approach to the problem is to eliminate x by picking a real value for x and then solving for c. Maybe you would like to start with  but this gives us  regardless the value of c. So how about ? It turns out to be

and

which leads to  and . Surprising result? I wonder how did the  come as it wouldn’t satisfy .

This is that rare occasion where letting x take a certain value doesn’t work out. A capable student could probably devise a method to eliminate the  as a solution but it is probably not worth the effort. Thus, I suggest solving the problem using the general method.