Domain of a Composite Function

jadewest · 2022-08-02 11:12:04

Hello,

I can't figure out what the domain for this exercise is for days. Can someone help me?

7 f(x) = 2x + 3 and g(x) = –x^2 + 5, find (g o g) (x).

(g o g) (x) = g (g(x))

= g (-x^2 + 5)

= (- (-x^2 + 5) ^2 + 5)

= (-(-x^2 + 5) (-x^2 + 5) + 5)

= (- (x^4 -10x^2 + 25) + 5)

= -x^4 + 10x^2 - 20

Thank you,
Jade

Bob · 2022-08-02 20:08:23

hi Jade,

When a function is defined it is the responsibility of the definer to say what domain applies as part of the definition, not the student. So I'm wondering why you are asking this.

Looking at the details of the question as given by you, firstly your algebra is correct. There are no features such as square roots that might limit what values can be entered into the function, so I don't see what is wrong with saying the domain could be all real numbers. But the question still 'works' if the domain is, for example, negative integers, or complex numbers.

What does it say at the start of the worksheet? Maybe there's a clue there.

The range is the set of possible outputs from the function. That set may be limited by the function's actions (eg. Sine outputs are in the range -1≤ y ≤ +1)

For gg I used the MIF function grapher to plot the function. It's a quartic so it has three turning points. The curve starts at negative ∞, rises to a maximum, tips over and drops to a minimum, then up to a second maximum, before dropping to negative infinity again. The two maxima are at the same level and zooming in, it appears to be at y = 5. There are calculus techniques that would allow me to confirm this but I don't think you've done this yet. So you could say that the range is (- ∞, +5] The graph confirms what I have said about the domain.

Try it yourself: https://www.mathsisfun.com/data/function-grapher.php

Bob

Math Is Fun Forum

#1 2022-08-02 11:12:04

Domain of a Composite Function

#2 2022-08-02 20:08:23

Re: Domain of a Composite Function

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