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Numerade Educator

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Problem 25 Medium Difficulty

If $ f(x) = 3x^2 - x + 2 $ , find $ f(2) $ , $ f(-2) $ , $ f(a) $ , $ f(-a) $ , $ f(a + 1) $ , 2 $ f(a) $ , $ f(2a) $ , $ f(a^2) $ , $ [ f(a) ]^2 $ , and $ f(a + h) $.

Answer

$f(x)=3 x^{2}-x+2$
$f(2)=3(2)^{2}-2+2=12-2+2=12$
$f(-2)=3(-2)^{2}-(-2)+2=12+2+2=16$
$f(a)=3 a^{2}-a+2$
$f(-a)=3(-a)^{2}-(-a)+2=3 a^{2}+a+2$
$f(a+1)=3(a+1)^{2}-(a+1)+2=3\left(a^{2}+2 a+1\right)-a-1+2=3 a^{2}+6 a+3-a+1=3 a^{2}+5 a+4$
$2 f(a)=2 \cdot f(a)=2\left(3 a^{2}-a+2\right)=6 a^{2}-2 a+4$
$f(2 a)=3(2 a)^{2}-(2 a)+2=3\left(4 a^{2}\right)-2 a+2=12 a^{2}-2 a+2$
$f\left(a^{2}\right)=3\left(a^{2}\right)^{2}-\left(a^{2}\right)+2=3\left(a^{4}\right)-a^{2}+2=3 a^{4}-a^{2}+2$
$\begin{aligned}[f(a)]^{2} &=\left[3 a^{2}-a+2\right]^{2}=\left(3 a^{2}-a+2\right)\left(3 a^{2}-a+2\right) \\ &=9 a^{4}-3 a^{3}+6 a^{2}-3 a^{3}+a^{2}-2 a+6 a^{2}-2 a+4=9 a^{4}-6 a^{3}+13 a^{2}-4 a+4 \\ f(a+h) &=3(a+h)^{2}-(a+h)+2=3\left(a^{2}+2 a h+h^{2}\right)-a-h+2=3 a^{2}+6 a h+3 h^{2}-a-h+2 \end{aligned}$

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Video Transcript

in this problem, we have f of X and were asked to find a whole bunch of different function values. And this is preparing us for things that are coming up later in the course. So we're going to start with f of two. So we substitute to infer X. We get three times two squared minus two plus two, and that's going to be three times for minus two plus two a zero. And that's just going to be 12. And now we're going to find F of negative, too. So we're going to substitute negative to and for X. So we have three times negative two squared, minus negative two plus two. And that gives us three times four plus two plus two. That's 12 plus two plus two. So that's 16. Next, we're going to find f of A. So we substitute A. And for X and F of A is simply three a squared minus a plus two. There's nothing more we can do with that. Now we're going to find F of the opposite of A and will substitute the opposite of a in for X so have three times the opposite of a squared minus the opposite of a plus two. We can simplify this. The opposite of a squared is going to be equivalent to a squared. So we have three. A squared plus a plus two. Now we're going to find F of A plus one. So we substitute a plus one. And for X, we have three times a quantity. A plus one squared, minus the quantity A plus one plus two. So we're going to need to multiply a plus. One quantity squared out using the foil method, and we get three times a quantity. A squared plus two A plus one, and we can distribute the minus sign in front of the quantity eight. Plus, once we get minus a minus one and we still have plus two at the end. So now we can distribute the three and we have three. A squared plus six a plus three minus a minus one plus two. Now let's combine our like terms and we have three a squared plus five a plus four. Okay, so we've done 12345 of them. We have 12345 more to go. All right, so now let's find two times f of a. So we already found that f of a waas three a squared minus a plus two. So two times that we're just going to double that and we get six a squared minus to a plus four. Now we're going to find f of two a c. These air different, and we need to know that these are not the same thing. Two times f of A is not the same as F of two way. There's an order of operations change going on here, So we're going to substitute to a and for X. So get three times to a squared minus two a plus two. And let's simplify that. So to a squared to square a square, that's gonna be for a squared times. Three. That will be 12 a squared minus to a plus two so we can see that is definitely not the same as the previous one. Now we want to find f of a squared, so we substitute a squared in the function for X. So it three times a squared B squared minus a squared plus tube, and that works out to be three a to the fourth minus a squared plus two. Okay, now we're finding f of a quantity squared. Now, again, this is going to be different than the one we just did. I'm going to do it on the next flight. However, you should note that the A squared inside is different than squaring on the outside. So expect the answer to be different. So we're doing f of a quantity squared. We already know that f of a is three x squared minus X plus two hoops three a squared minus a plus two. Now we need to square that. Okay, Squaring it means multiplying that whole thing by itself. So we have a try. No meal that we're going to multiply by a try. No meal. We need to be very systematic about this to make sure we don't miss anything. So I'm gold going to multiply three a squared by all three of these. And that gives me 98 to the fourth power minus three. A cubed plus six a squared. And then I'm going to multiply the opposite of a times all three things and that gives me minus three a cubed plus a squared minus two A and Now I'm going to multiply two by all of these three things, and that gives me six a squared minus to a plus four. Now we'll combine our like terms to have 98 of the fourth power, and we have minus three a cubed and another minus three a cube, so that would be minus six a cubed. And we have six a squared and plus six a squared and plus another Sorry six, a squared plus one, a squared plus another six a squared and that gives us 13 a squared. And then we have minus two a and minus another two way. So that gives us minus for a and then we have a four. All right, so it's very different from the previous just squaring the A first. And finally we have f of a plus h. So again f of X is three x squared minus X plus two. So f of a plus h is what we get when we substitute a plus h in for X. This is foreshadowing two things you're going to be doing later. So what we need to dio just like we did on a previous one where we had a plus one, and there is We need to multiply this out using the foil method so three times a quantity a squared plus two, a h plus h squared and then we can distribute the minus sign and we have minus a minus H and plus two. Now we can distribute the three. So three a squared plus six a H plus three age squared minus a minus h plus two. And it doesn't look like any of those air like terms, so that's a Sfar as we can go with that.