Discrepancy SG 3 v4, equations inequalities and VICs ch5, Q6

[sscolley]

Hi,

There seems to be a discrepancy between the solution method given for Q6 on page 81 and the worked example on page 73 for compound functions - to find the value for x for which f(g(x)) = g(f(x)).

The two methods yield different answers. It seems to me that the method on p73 is more appropriate.

Please help. I test in three days.

Best,
Simon

[zdgura]

Simon function question that's also deals with a quadratic. As a result their are two different answers. Set f(x) and g(x) equal to each other and then solve. Maybe you are getting confused w/the whole f(x) = and equation that also has "x's" in it. As a rule of thumb I just ignore this. In fact to make it visually easier make f(x) = A, where A equals the given equation. Then make g(x) = b (again equaling the corresponding equation). Then you note that A = B. Substitute the correction for A & B and then solve. Maybe I'm completely misunderstanding your question, but I hope that helps.

P.S. Good luck on your test!

[jnelson0612]

Hi,

There seems to be a discrepancy between the solution method given for Q6 on page 81 and the worked example on page 73 for compound functions - to find the value for x for which f(g(x)) = g(f(x)).

The two methods yield different answers. It seems to me that the method on p73 is more appropriate.

Please help. I test in three days.

Best,
Simon

Hi Simon,
I hope your test went well!

As the previous poster noted, these are two different questions. EIVs book page 79 #6 asks for x values that allow f(x)=g(x). f(x)=2x^2 - 4 and g(x)=2x. Thus, we want 2x^2 - 4 to equal 2x; we can set them equal and solve.

For the compound function on page 73, we are told that:
f(x)=x^3+1
g(x)=2x

We want to know for what value does f(g(x))=g(f(x)).

This is tricky. Let's start with the left side. What is f(g(x))? Well, in this case we need to replace g(x) with what it is defined as: 2x. So the left side is f(2x). Now, we take 2x and substitute it for x everywhere in the f(x) function. Thus, f(2x)=(2x)^3 + 1, or 8x^3+1.

Now let's tackle the right side: g(f(x)). What is f(x)? It is x^3+1. So we are figuring out g(x^3+1). g(x)=2x, so 2(x^3+1) = 2x^3 + 2.

So now we have 8x^3+1 = 2x^3 + 2, which yields 6x^3 = 1. Divide by 6 on both sides, then take the cube root of both sides, and x equals the cube root of 1/6.

I hope this helps clarify things a bit; these are two different questions and should be dealt with differently.

[rexbrown]

"Now let's tackle the right side: g(f(x)). What is f(x)? It is x^3+1. So we are figuring out g(x^3+1). g(x)=2x, so 2(x^3+1) = 2x^3 + 2."

Why is it 2(x^3+1) and not 2x(^3+1) ? I know the math makes sense but for future reference I'd like to know why the left side became 8x^3 and the right side is 2x^3?

[RonPurewal]

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