TMUA Master

A function problem does not always give us a formula such as $f(x)=\cdots$ Instead, we may be told how particular inputs and outputs are related. In these problems, finding the complete formula for the function may be unnecessary.

The useful habit is to follow the information that the problem gives us.

Start with the Input You Need

Suppose $f(3x-1)=x^2+2x$ for all real $x$, and we want to find $f(8)$.

The expression $3x-1$ is the input of the function. To obtain $f(8)$, we therefore need $3x-1=8$.

This gives $x=3$. We can now return directly to the given relationship: $f(8)=3^2+2(3)=15$.

There was no need to find a formula for $f(x)$.

The reasoning can be viewed as a short chain:

More generally, if $f(A(x))=B(x)$, then $A(x)$ is the input being sent into $f$, while $B(x)$ is the corresponding output.

To find $f(c)$, the natural first question is $A(x)=c$.

Once we know which values of $x$ produce the required input, we can use the corresponding value of $B(x)$.

Do not find the whole function unless you need it. First match the input you want to the input you have been given.

Follow the Intermediate Value

The same idea applies to composite functions.

Suppose the only solutions of $f(t)=4$ are $t=-1 \quad \text{and} \quad t=3$.

Now suppose $f(g(x))=4$.

Think of the information as passing through two stages: $x\longrightarrow g(x)\longrightarrow f(g(x))$.

We already know which inputs make $f$ produce the output 4. Therefore the intermediate value $g(x)$ must be one of them: $g(x)=-1 \quad \text{or} \quad g(x)=3$.

We do not yet need to solve for $x$. The first step is simply to identify what $g(x)$ is allowed to be.

This is often a better way to read a composition than immediately trying to expand it.

One Input Must Have One Output

There is one important condition hidden inside every function relationship:

The same input cannot produce two different outputs.

Consider $f(x^2+1)=2x^2-3$.

Suppose we want $f(10)$. Matching the input gives $x^2+1=10$, so $x=3 \quad \text{or} \quad x=-3$.

Both values produce the same input, 10. They also produce the same output because the right-hand side depends only on $x^2$: $2x^2-3=15$.

So there is no conflict: $f(10)=15$.

Now compare this with $f(x^2)=x+4$.

If this were true for every real $x$, consider $x=2$ and $x=-2$.

Both values give the same input to $f$, but they would require $f(4)=6$ and $f(4)=2$.

That is impossible. A function cannot assign two different outputs to the same input.

So function relationships can sometimes tell us whether a proposed condition is possible, even when we know nothing else about the function.

A Useful Way to Read Function Problems

When you meet an unfamiliar function relationship, separate the problem into three parts:

Then check one more thing whenever several values lead to the same function input:

Do they also lead to the same output?

This gives a useful progression:

$$\text{required value} \longrightarrow \text{match the input} \longrightarrow \text{follow the relationship} \longrightarrow \text{keep all valid possibilities}.$$

Key Idea: Read a function as a relationship between inputs and outputs. Follow only the information you need, and when an intermediate input has several possibilities, keep them all until the conditions allow you to decide between them.