When you're given the definition of a function as you are here, \(\displaystyle f(x)=x^2-x\), your job to calculate a function is to take the value in parentheses and plug that in for \(\displaystyle x\) wherever it appears in the definition. Here, qualitatively, you're being told "whatever \(\displaystyle x\) is, square it and then subtract \(\displaystyle x\) from that square."  That means that:

\(\displaystyle f(9)=9^2-9\)

So:

\(\displaystyle f(9)=81-9=72\)

And for \(\displaystyle f(7)\) you'd have:

\(\displaystyle f(7)=7^2-7\)

So:

\(\displaystyle f(7)=49-7=42\)

This means that \(\displaystyle f(9)-f(7)=72-42=30\), so the correct answer is \(\displaystyle 30\).

Whenever you're working with a function \(\displaystyle f(x)=\) with an algebraic definition like the one you're given, your job is to plug in the value (in this case \(\displaystyle -1\)) wherever \(\displaystyle x\) appears in that definition.  This means that your work should look like:

\(\displaystyle f(-1)=\frac{(-1)^3-2(-1)^2+3(-1)+1)}{-1}\)

The key here becomes keeping the negative/positive signs in place given all the addition/subtraction and multiplication of negatives in the numerator. Once you've applied the exponents and multiplication in the numerator, you should have:

\(\displaystyle \frac{-1-2-3+1}{-1}\)

This then simplifies to \(\displaystyle \frac{-5}{-1}\), in which the negatives in numerator and denominator cancel, leaving you with just \(\displaystyle 5\) as your answer.

When you're given a function definition in the form \(\displaystyle f(x)=\)... as you are here, your job is then to plug in the value in parentheses anywhere that \(\displaystyle x\) appears. That means that to solve for \(\displaystyle f(3)\) you'll just plug in \(\displaystyle 3\) for the \(\displaystyle x\) in \(\displaystyle 2x+1\):

\(\displaystyle f(3)=2(3)+1=7\)

And to solve for \(\displaystyle f(4)\) you would do the same thing, plugging in \(\displaystyle 4\) in place of \(\displaystyle x:\)

\(\displaystyle f(4)=2(4)+1=9\)

To finish the problem, you'll then add 7 + 9 to get the correct answer, 16.

This problem adds a twist to the classic function setup. You should know that whenever you're dealing with a function defined as \(\displaystyle f(x)=...\), your job is to plug in the given input value anywhere there's an \(\displaystyle x\) term. Often that value is a number, but here the next thing you're told is that \(\displaystyle f(x+y)=3x^2+24x+49\), meaning that your input value is \(\displaystyle x+y\). 

The steps remain the same, however: just plug in \(\displaystyle x+y\) to \(\displaystyle f(x)=3x^2+1\) and you can solve:

This means that \(\displaystyle f(x+y)=3(x+y)^2+1\), and you know that it should expand to \(\displaystyle 3x^2+24x+49\). So you can set up an equation and solve:

\(\displaystyle 3(x+y)^2+1=3x^2+24x+49\)

Square the parenthetical term to get:

\(\displaystyle 3(x^2+2xy+y^2)+1=3x^2+24x+49\)

And then distribute the 3:

\(\displaystyle 3x^2+6xy+3y^2+1=3x^2+24x+49\)

And you can now subtract \(\displaystyle 3x^2\) and \(\displaystyle 1\) from each side to simplify:

\(\displaystyle 6xy + 3y^2=24x+48\)

When you divide both sides by 3 the answer should start to become clear:

\(\displaystyle 2xy+y^2=8x+16\)

You cannot get rid of the \(\displaystyle x\) term so you can see that \(\displaystyle 2xy=8x\) and \(\displaystyle y^2=16\).  So \(\displaystyle y=4\) satisfies both terms, making \(\displaystyle 4\) the correct answer.

Whenever you're working with a function defined as \(\displaystyle f(x)=...\), your job is to take the input value--the value in parentheses--and insert it wherever there's an \(\displaystyle x\) in the definition. Since here \(\displaystyle f(4)\) tells you that your input value is \(\displaystyle 4\), you can plug that into the function:

\(\displaystyle f(4)=50-a(4)^2\)

You're told that \(\displaystyle f(4)=2\) so you can set up an equation:

\(\displaystyle 50-a(4)^2=2\)

And then you can perform the exponent:

\(\displaystyle 50-16a=2\)

And work to get like terms together:

\(\displaystyle 50-2=16a\)

So \(\displaystyle 48=16a\), meaning that \(\displaystyle a=3\).

The question gives you that \(\displaystyle f(x+2)=0\) and asks for the value of \(\displaystyle x\). The easiest thing to start here is to find the value for \(\displaystyle t\) where \(\displaystyle f(t)=0\).

You're given that \(\displaystyle f(t)=(t-4)^2(t-7)\). You can then set this equal to 0 to get \(\displaystyle 0=(t-4)^2(t-7)\). This means that \(\displaystyle t\) has to equal either \(\displaystyle 4\) or \(\displaystyle 7\).  In order for \(\displaystyle x+2\) to equal either of these, \(\displaystyle x\) must be \(\displaystyle 2\) or \(\displaystyle 5\). Only \(\displaystyle 5\) is a provided answer choice, so \(\displaystyle 5\) is the correct answer.

When you're working with "nested functions" - problems in which you're asked to apply a function to a function, such as \(\displaystyle f(f(3))\) here - you should follow classic Order-of-Operations and start with the interior parentheses first. Here that means taking \(\displaystyle f(3)\), the inner function, and then using that result as your input for the outer function.

In the definition \(\displaystyle f(x)=x^{2}-1\), the left-hand side of the equation defines your "input" saying "whatever you see in the parentheses where \(\displaystyle x\) currently is, do to that value what is done to \(\displaystyle x\) on the right-hand side of the equation." And then the right-hand side of the equation tells you what to do to your input. Here it's saying "take your input and square it, then subtract one."

When you apply that to \(\displaystyle f(3)\), you'll do exactly what that says: plug \(\displaystyle 3\) into the \(\displaystyle x\) spots, meaning you'll take \(\displaystyle 3^{2}-1\). The result of that is \(\displaystyle 8\). So \(\displaystyle f(f(3))=f(8)\).

Now with \(\displaystyle f(8)\) your input is \(\displaystyle 8\), so you'll plug in \(\displaystyle 8\) for \(\displaystyle x\) in \(\displaystyle f(x)=x^{2}-1\). This means that you'll have:

\(\displaystyle f(8)=8^{2}-1\)

That simplifies to \(\displaystyle 64-1=63\).

This problem tests your familiarity and comfort with function notation. When you're given a definition like \(\displaystyle f(x)=100-2^{x}\), it's important to recognize that \(\displaystyle x\) is the "input" (whatever they tell you \(\displaystyle x\) is, you then put that into the equation), and that \(\displaystyle f(x)\) is the "output" (once you've put your input through the equation, the result is \(\displaystyle f(x)\).

Here you're given the function definition and then told that the output, \(\displaystyle f(x)\), is equal to \(\displaystyle 6\). Then you're asked to solve for \(\displaystyle x\), which means that you're asked to solve for the input.

So what they're really asking is "for what number, when you take \(\displaystyle 2\) to that power and then subtract the result from \(\displaystyle 100\), would you end up with \(\displaystyle 6\)?"

In equation form, that's \(\displaystyle 6=100-2^{x}\). Performing the algebra, you can add \(\displaystyle 2^{x}\) to each side and subtract \(\displaystyle 6\) from each side.

That gives you: \(\displaystyle 2^{x}=94\). Knowing your powers of \(\displaystyle 2\), you should recognize that \(\displaystyle 2^{6}=64\) and \(\displaystyle 2^{7}=128\), so \(\displaystyle x\) must be between \(\displaystyle 66\) and \(\displaystyle 77\).

To solve, you can simply set the output of the function \(\displaystyle \frac{2}{3}\) equal to the algebraic expression: 

\(\displaystyle \frac{2}{3}=\frac{x}{(1-x)}\)

And cross-multiply:

\(\displaystyle 2-2x=3x\)

And solve for \(\displaystyle x\):

\(\displaystyle 2=5x\)

\(\displaystyle \frac{2}{5}=x\)

Alternatively, you could have recognized that we have a positive result, so the numerator of our fraction cannot be \(\displaystyle 0\), and that the values for x greater than 1 would all leave a negative denominator, so that cannot be, either.

If \(\displaystyle f(x)=g(x)\), you can simply set the two items equal: \(\displaystyle x^{2}-9=16x-73\). From there, you can "complete the square" by subtracting the right-hand side and moving it to the left, getting to: \(\displaystyle x^{2}-16x+64=0\). This should look familiar as a common algebraic equation; it factors to \(\displaystyle (x-8)^{2}=0\). Accordingly, the solution must be \(\displaystyle x=8\).