# If $f(x)=2x-5$, when $f(x)=13$

If $\ f(x) = 2x - 5 \$ , find $\ x \$ when $\ f(x) = 13 \$. I substituted 13 and got 21, but the answer was 9?

Can someone please show me working out for this type of question I'm bad at math!

We know that $f(x)=13$. This means that: $$f(x)=13=2x-5$$ $$\implies 13=2x-5$$ Now all we need to do is to solve this equation. First add $5$ to both sides. $$13+5=2x$$ $$18=2x$$ Now divide by $2$ on both sides. $$\frac{18}2=x$$ $$9=x$$ That is why $x=9$, as you said above.

If you want a more deeper understanding of what this problem is asking you to do, read on.

Say I have the function $f(x)=2x-1$, and I want to find $x$ when $f(x)=1$. This is like the reverse of plugging in an $x$ value into the function. We want to find what $x$ value we have to plug in.

What I will do now is find the inverse of $f(x)$. The inverse of any function $f(x)$ (denoted as $f^{-1}(x)$) must follow this rule (provided domain is correct): $$f^{-1}(f(x))=x$$ This may seem very confusing to you. What this means is that the inverse function $f^{-1}(x)$ must have an $x$ value of $f(x)$. You will understand later.

How do you find inverses of functions? First, you substitute $y$ for $f(x)$. $$y=2x-1$$ Then you switch $x$ and $y$. $$x=2y-1$$ Now you have to solve for $y$ in terms of $x$. $$x+1=2y$$ $$y=\frac{x+1}2$$ Lastly, substitute $f^{-1}(x)$ for $y$. $$f^{-1}(x)=\frac{x+1}2$$ That is the inverse of $f(x)$. Now I will show you what I mean by $f^{-1}(f(x))=x$. We evaluate $f(x)$ first, then plug in the value of $f(x)$ to $f^{-1}(x)$. Let's plug in an $x$ value of $3$. First, let's find $f(3)$. $$f(3)=2(3)-1=6-1=5$$ Now we plug in $5$ as our $x$ value in $f^{-1}(x)$. $$f^{-1}(5)=\frac{5+1}2=\frac 62=3$$ That means $f^{-1}(f(3))=3$, just as the rule says. This is a good way to check if your inverse is correct.

Since $f^{-1}(x)$ is the opposite of $f(x)$, then that means if $f(x)=y$, then $f^{-1}(y)=x$. This means that $f^{-1}(1)$ equals the $x$ value we have been trying to find. $$f^{-1}(1)=\frac{1+1}2=\frac 22=1$$ Therefore our $x$ value is $1$. Let's see if this is correct. $$f(1)=2(1)-1=2-1=1 \ \ \checkmark$$ You can use this way to find $x$ in your problem if you want. But the way I showed you at the top of the answer is IMHO the best way, because it is quicker.

Hope I helped

$$f(x) = 2x-5$$

If $f(x)=13$, then that means you want to find an $x$, so that when you plug it into $f(x)$, you get 13.

$$f(x) = 13\\2x-5=13\\2x=18\\x=9$$

You don't want to find $f(13)$ which is what you're doing by substituting $13$ for $x$ (and why you're getting $21$). You want $x$ such that $f(x) = 13$ which means $2x-5 = 13$. Can you take it from here?