Math / Algebra

QuestionState the domain and range of the function. $f(x)=\log _{7}(x+1)$
Domain: $\square$ (Enter your answer in interval notation.)
Range: $\square$ (Enter your answer in interval notation.)

Studdy Solution

STEP 1

What is this asking? We need to find all the valid inputs ( $x$ ) and outputs ( $f(x)$ ) for the logarithmic function $f(x) = \log_7(x+1)$ . Watch out! Remember that we can only take the logarithm of a positive number!

STEP 2

1. Find the domain.
2. Find the range.

STEP 3

Alright, let's find the domain!
The domain of a function is all the possible $x$ values we can plug in.
Since we're dealing with a logarithm, the argument inside the logarithm must be greater than zero.
In our case, the argument is $x + 1$ .

STEP 4

So, we need $x + 1 > 0$ .
To isolate $x$ , we'll subtract $1$ from both sides of the inequality.
This gives us $x + 1 - 1 > 0 - 1$ , which simplifies to $x > -1$ .

STEP 5

Now, we write this in interval notation.
Since $x$ is greater than $-1$ , our interval starts at $-1$ and goes to infinity.
Because $x$ cannot equal $-1$ , we use a parenthesis to show that $-1$ is not included.
Infinity always gets a parenthesis.
So, our domain is $(-1, \infty)$ .

STEP 6

Now for the range!
The range is all the possible output values, which is $f(x)$ in our case.
Logarithms can output any real number.
Think about it: If we keep making $x$ bigger and bigger, $\log_7(x+1)$ will also get bigger and bigger.
And if we make $x$ get closer and closer to $-1$ (without actually reaching it!), $\log_7(x+1)$ will become a very large negative number!

STEP 7

So, the range of our function is all real numbers, which we write in interval notation as $(-\infty, \infty)$ .

STEP 8

Domain: $(-1, \infty)$
Range: $(-\infty, \infty)$

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QuestionState the domain and range of the function. $f(x)=\log _{7}(x+1)$
Domain: $\square$ (Enter your answer in interval notation.)
Range: $\square$ (Enter your answer in interval notation.)

Studdy Solution

STEP 1

What is this asking? We need to find all the valid inputs ( $x$ ) and outputs ( $f(x)$ ) for the logarithmic function $f(x) = \log_7(x+1)$ . Watch out! Remember that we can only take the logarithm of a positive number!

STEP 2

1. Find the domain.
2. Find the range.

STEP 3

Alright, let's find the domain!
The domain of a function is all the possible $x$ values we can plug in.
Since we're dealing with a logarithm, the argument inside the logarithm must be greater than zero.
In our case, the argument is $x + 1$ .

STEP 4

So, we need $x + 1 > 0$ .
To isolate $x$ , we'll subtract $1$ from both sides of the inequality.
This gives us $x + 1 - 1 > 0 - 1$ , which simplifies to $x > -1$ .

STEP 5

Now, we write this in interval notation.
Since $x$ is greater than $-1$ , our interval starts at $-1$ and goes to infinity.
Because $x$ cannot equal $-1$ , we use a parenthesis to show that $-1$ is not included.
Infinity always gets a parenthesis.
So, our domain is $(-1, \infty)$ .

STEP 6

STEP 7

So, the range of our function is all real numbers, which we write in interval notation as $(-\infty, \infty)$ .

STEP 8

Domain: $(-1, \infty)$
Range: $(-\infty, \infty)$

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QuestionState the domain and range of the function. f(x)=log⁡7(x+1)f(x)=\log _{7}(x+1)f(x)=log7​(x+1)Domain: □\square□ (Enter your answer in interval notation.)Range: □\square□ (Enter your answer in interval notation.)

Studdy Solution

STEP 1

What is this asking? We need to find all the valid inputs (xxx) and outputs (f(x)f(x)f(x)) for the logarithmic function f(x)=log⁡7(x+1)f(x) = \log_7(x+1)f(x)=log7​(x+1). Watch out! Remember that we can only take the logarithm of a *positive* number!

STEP 2

1. Find the domain.2. Find the range.

STEP 3

Alright, let's **find the domain**!The domain of a function is all the possible xxx values we can plug in.Since we're dealing with a logarithm, the argument *inside* the logarithm *must* be greater than zero.In our case, the argument is x+1x + 1x+1.

STEP 4

So, we need x+1>0x + 1 > 0x+1>0.To **isolate** xxx, we'll subtract 111 from both sides of the inequality.This gives us x+1−1>0−1x + 1 - 1 > 0 - 1x+1−1>0−1, which simplifies to x>−1x > -1x>−1.

STEP 5

STEP 6

STEP 7

So, the range of our function is all real numbers, which we write in interval notation as (−∞,∞)(-\infty, \infty)(−∞,∞).

STEP 8

Domain: (−1,∞)(-1, \infty)(−1,∞)Range: (−∞,∞)(-\infty, \infty)(−∞,∞)

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionState the domain and range of the function. $f(x)=\log _{7}(x+1)$
Domain: $\square$ (Enter your answer in interval notation.)
Range: $\square$ (Enter your answer in interval notation.)

What is this asking? We need to find all the valid inputs ( $x$ ) and outputs ( $f(x)$ ) for the logarithmic function $f(x) = \log_7(x+1)$ . Watch out! Remember that we can only take the logarithm of a positive number!

1. Find the domain.
2. Find the range.

Alright, let's find the domain!
The domain of a function is all the possible $x$ values we can plug in.
Since we're dealing with a logarithm, the argument inside the logarithm must be greater than zero.
In our case, the argument is $x + 1$ .

So, we need $x + 1 > 0$ .
To isolate $x$ , we'll subtract $1$ from both sides of the inequality.
This gives us $x + 1 - 1 > 0 - 1$ , which simplifies to $x > -1$ .

So, the range of our function is all real numbers, which we write in interval notation as $(-\infty, \infty)$ .

Domain: $(-1, \infty)$
Range: $(-\infty, \infty)$