Math / Algebra

Question25. Find the domain of the function $f(x)$ . $f(x)=\sqrt{1-2 x}$

Studdy Solution

STEP 1

What is this asking? We need to find all the allowed $x$ values for the function $f(x) = \sqrt{1 - 2x}$ . Watch out! We can't take the square root of a negative number!

STEP 2

1. Set up the inequality
2. Isolate $x$

STEP 3

Alright, so we've got this square root, and we know that whatever is inside that square root can't be negative.
It can be zero, or it can be positive, but it can't be negative!
So, what's inside?
It's $1 - 2x$ .

STEP 4

That means we can write this little inequality: $1 - 2x \ge 0$ This just says "one minus two x is greater than or equal to zero".
This is exactly what we want because we need the stuff inside the square root to be zero or positive!

STEP 5

Now, let's isolate $x$ .
First, we'll subtract 1 from both sides of our inequality.
Remember, what we do to one side, we must do to the other! $1 - 2x - 1 \ge 0 - 1$ $-2x \ge -1$

STEP 6

Now, we need to get $x$ all by itself.
We're going to divide both sides by $-2$ . But here's the catch! When we divide or multiply an inequality by a negative number, we have to flip the inequality sign.
So, $\ge$ becomes $\le$ .
Let's do it! $\frac{-2x}{-2} \le \frac{-1}{-2}$ $x \le \frac{1}{2}$

STEP 7

So, the domain of our function $f(x)$ is all $x$ values less than or equal to $\frac{1}{2}$ .
We can write this neatly as: $x \le \frac{1}{2}$ or using interval notation: $\left(-\infty, \frac{1}{2}\right]$

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Question25. Find the domain of the function $f(x)$ . $f(x)=\sqrt{1-2 x}$

Studdy Solution

STEP 1

What is this asking? We need to find all the allowed $x$ values for the function $f(x) = \sqrt{1 - 2x}$ . Watch out! We can't take the square root of a negative number!

STEP 2

1. Set up the inequality
2. Isolate $x$

STEP 3

Alright, so we've got this square root, and we know that whatever is inside that square root can't be negative.
It can be zero, or it can be positive, but it can't be negative!
So, what's inside?
It's $1 - 2x$ .

STEP 4

That means we can write this little inequality: $1 - 2x \ge 0$ This just says "one minus two x is greater than or equal to zero".
This is exactly what we want because we need the stuff inside the square root to be zero or positive!

STEP 5

Now, let's isolate $x$ .
First, we'll subtract 1 from both sides of our inequality.
Remember, what we do to one side, we must do to the other! $1 - 2x - 1 \ge 0 - 1$ $-2x \ge -1$

STEP 6

Now, we need to get $x$ all by itself.
We're going to divide both sides by $-2$ . But here's the catch! When we divide or multiply an inequality by a negative number, we have to flip the inequality sign.
So, $\ge$ becomes $\le$ .
Let's do it! $\frac{-2x}{-2} \le \frac{-1}{-2}$ $x \le \frac{1}{2}$

STEP 7

So, the domain of our function $f(x)$ is all $x$ values less than or equal to $\frac{1}{2}$ .
We can write this neatly as: $x \le \frac{1}{2}$ or using interval notation: $\left(-\infty, \frac{1}{2}\right]$

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Question25. Find the domain of the function f(x)f(x)f(x). f(x)=1−2xf(x)=\sqrt{1-2 x}f(x)=1−2x​

Studdy Solution

STEP 1

What is this asking? We need to find all the allowed xxx values for the function f(x)=1−2xf(x) = \sqrt{1 - 2x}f(x)=1−2x​. Watch out! We can't take the square root of a negative number!

STEP 2

1. Set up the inequality2. Isolate xxx

STEP 3

Alright, so we've got this square root, and we know that whatever is *inside* that square root can't be negative.It can be zero, or it can be positive, but it *can't* be negative!So, what's inside?It's 1−2x1 - 2x1−2x.

STEP 4

That means we can write this little inequality: 1−2x≥01 - 2x \ge 01−2x≥0 This just says "one minus two x is greater than or equal to zero".This is exactly what we want because we need the stuff *inside* the square root to be zero or positive!

STEP 5

Now, let's **isolate** xxx.First, we'll subtract 1 from both sides of our inequality.Remember, what we do to one side, we *must* do to the other! 1−2x−1≥0−11 - 2x - 1 \ge 0 - 11−2x−1≥0−1 −2x≥−1-2x \ge -1−2x≥−1

STEP 6

STEP 7

So, the **domain** of our function f(x)f(x)f(x) is all xxx values less than or equal to 12\frac{1}{2}21​.We can write this neatly as: x≤12x \le \frac{1}{2}x≤21​ or using interval notation: (−∞,12]\left(-\infty, \frac{1}{2}\right](−∞,21​]

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Question25. Find the domain of the function $f(x)$ . $f(x)=\sqrt{1-2 x}$

What is this asking? We need to find all the allowed $x$ values for the function $f(x) = \sqrt{1 - 2x}$ . Watch out! We can't take the square root of a negative number!

1. Set up the inequality
2. Isolate $x$

Alright, so we've got this square root, and we know that whatever is inside that square root can't be negative.
It can be zero, or it can be positive, but it can't be negative!
So, what's inside?
It's $1 - 2x$ .

That means we can write this little inequality: $1 - 2x \ge 0$ This just says "one minus two x is greater than or equal to zero".
This is exactly what we want because we need the stuff inside the square root to be zero or positive!

Now, let's isolate $x$ .
First, we'll subtract 1 from both sides of our inequality.
Remember, what we do to one side, we must do to the other! $1 - 2x - 1 \ge 0 - 1$ $-2x \ge -1$

So, the domain of our function $f(x)$ is all $x$ values less than or equal to $\frac{1}{2}$ .
We can write this neatly as: $x \le \frac{1}{2}$ or using interval notation: $\left(-\infty, \frac{1}{2}\right]$