For the following question :
$f(x) = 2x^2 + 4x $
It asks me to express the following in terms of $x$:
$f(-2x)$
What does the question mean by this?
- Does it mean make $x$ the subject?
4 Answers
$\begingroup$Good question. This is a phrase mathematicians and mathematics teachers use a lot, and it has a specific meaning that isn't entirely clear to the learner.
Idiomatically speaking, to write a function “in terms of” a given variable or variables means to write an algebraic expression using only that variable or variables.
So for instance, given an equation $x+2y-3z = 0$, we can solve for $z$ in terms of $x$ and $y$ as $z=\frac{1}{3}(x+2y)$.
Literally speaking, terms are the pieces that compose an expression. So in the expression $8x^2-8x$, $8x^2$ and $8x$ are terms combined by the subtraction function. The expression is in terms of $x$ since each term in the expression has only the variable $x$ (and constants) in it.
$\endgroup$ $\begingroup$When it means
express in terms of $x$
It means to express the quantity you're finding in terms of $x$, the variable.
Therefore,
Since:
$$f(x) = 2x^2 + 4x$$
So,
$$f(-2x) = 2(-2x)^2 + 4(-2x) = 8x^2 - 8x$$
$\endgroup$ $\begingroup$It means find the function $g(x) = f(-2x)$ in such a way that everyobody who knows math can simply plug in any value of $x$ to find $g(x)$.
For example, if $f(x) = \sin(x)$, then $f(-2x) = \sin(-2x)$, or even better (always simplify if that is possible!) $\sin(-2x)=-\sin(2x)$ is the expression you are looking for.
$\endgroup$ $\begingroup$To evaluate $f(-2x)$, you will first compute $x'=-2x$, then $2x'^2+4x'$.
You are asked to remove the intermediate substitution step and come up with a straight expression $g(x)=f(-2x)$.
Obviously, $g(x)=f(-2x)=2(-2x)^2+4(-2x)=8x^2-8x$, which is the answer.
$\endgroup$
