The function f is defined by f(x) = 2x - 1 for x < 3 and f(x) = x^2… Step by Step

Step by step

The question

Question: find f(3) for f(x) = 2x − 1 if x < 3, x² if x ≥ 3.

f(x) = 2x − 1 if x < 3

There are four choices.

First we solve it, the normal way.

Then we pick.

x² if x ≥ 3

This is a piecewise function.

It has 2 rules.

Which rule we use depends on x.

Each x uses only one rule.

piecewise: a different rule for different x's

each x uses only one rule

1 Find the right piece

Let's see where each rule works on a number line.

So we use this rule: x².

Careful!

3 is an edge: two rules meet there.

It belongs to the rule with the "or equal to" sign.

Careful: x = 3 goes where the "or equal to" is

2 Put 3 in

Put 3 in place of x in x².

f(3) = 3²

3² means 3 times 3.

3² = 3 × 3

= 9

That is 9.

f(3) = 9

So f(3) = 9.

3 Check

Let's check: does 3 fit each rule's x's?

Only one must say yes.

first rule: 3 < 3 no ✗

second rule: 3 ≥ 3 yes

Only the second rule fits.

So f(3) is right!

4 Pick the choice

Our answer is f(3) = 9.

That is choice C.

C) 9

5 Why not the others

A is 5.

Wrong: used 2x − 1: that rule is only for x less than 3.

That is the Careful mistake from before.

A) 5 ✗ used 2x − 1: that rule is only for x less than 3

B is 6.

Wrong: 3 × 2: x² is 3 × 3.

B) 6 ✗ 3 × 2: x² is 3 × 3

D is 14.

Wrong: added both rules: 5 + 9.

D) 14 ✗ added both rules: 5 + 9

Let's look back.

First: find the right piece.

Then: put 3 in.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer