Math / Algebra

QuestionWhat are the domain and range of $f(x)=\log (x+6)-4$ ? domain: $x>-6$ ; range: $y>4$ domain: $x>-6$ ; range: all real numbers domain: $x>6$ ; range: $y>-4$ domain: $x>6$ ; range: all real numbers

Studdy Solution

STEP 1

What is this asking? We need to find the allowed $x$ values (domain) and the possible $y$ values (range) of the function $f(x) = \log(x+6) - 4$ . 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 valid inputs, or $x$ values, we can plug in.
Since we're dealing with a logarithm, the argument inside the logarithm must be greater than zero.
In this case, the argument is $x+6$ .

STEP 4

So, we need $x+6 > 0$ .
To isolate $x$ , we'll subtract 6 from both sides of the inequality: $x + 6 - 6 > 0 - 6$ .
This simplifies to $x > -6$ .
Boom! That's our domain: $x > -6$ .

STEP 5

Now, let's tackle the range!
The range is all the possible outputs, or $y$ values, our function can produce.
Think about the graph of $y = \log(x)$ .
It extends infinitely in both the positive and negative $y$ directions!

STEP 6

Our function is $f(x) = \log(x+6) - 4$ .
The $-4$ just shifts the graph down by 4 units.
Shifting the graph up or down doesn't change the fact that the logarithm can still reach any $y$ value.

STEP 7

Therefore, the range of our function is all real numbers.
In fancy math terms, we write that as $-\infty < y < \infty$ .

STEP 8

The domain of $f(x) = \log(x+6) - 4$ is $x > -6$ , and the range is all real numbers.

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QuestionWhat are the domain and range of $f(x)=\log (x+6)-4$ ? domain: $x>-6$ ; range: $y>4$ domain: $x>-6$ ; range: all real numbers domain: $x>6$ ; range: $y>-4$ domain: $x>6$ ; range: all real numbers

Studdy Solution

STEP 1

What is this asking? We need to find the allowed $x$ values (domain) and the possible $y$ values (range) of the function $f(x) = \log(x+6) - 4$ . 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 valid inputs, or $x$ values, we can plug in.
Since we're dealing with a logarithm, the argument inside the logarithm must be greater than zero.
In this case, the argument is $x+6$ .

STEP 4

So, we need $x+6 > 0$ .
To isolate $x$ , we'll subtract 6 from both sides of the inequality: $x + 6 - 6 > 0 - 6$ .
This simplifies to $x > -6$ .
Boom! That's our domain: $x > -6$ .

STEP 5

Now, let's tackle the range!
The range is all the possible outputs, or $y$ values, our function can produce.
Think about the graph of $y = \log(x)$ .
It extends infinitely in both the positive and negative $y$ directions!

STEP 6

Our function is $f(x) = \log(x+6) - 4$ .
The $-4$ just shifts the graph down by 4 units.
Shifting the graph up or down doesn't change the fact that the logarithm can still reach any $y$ value.

STEP 7

Therefore, the range of our function is all real numbers.
In fancy math terms, we write that as $-\infty < y < \infty$ .

STEP 8

The domain of $f(x) = \log(x+6) - 4$ is $x > -6$ , and the range is all real numbers.

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QuestionWhat are the domain and range of f(x)=log⁡(x+6)−4f(x)=\log (x+6)-4f(x)=log(x+6)−4 ? domain: x>−6x>-6x>−6; range: y>4y>4y>4 domain: x>−6x>-6x>−6; range: all real numbers domain: x>6x>6x>6; range: y>−4y>-4y>−4 domain: x>6x>6x>6; range: all real numbers

Studdy Solution

STEP 1

What is this asking? We need to find the allowed xxx values (domain) and the possible yyy values (range) of the function f(x)=log⁡(x+6)−4f(x) = \log(x+6) - 4f(x)=log(x+6)−4. Watch out! Remember that we can only take the logarithm of a *positive* number!

STEP 2

1. Find the Domain2. Find the Range

STEP 3

Alright, let's **find the domain**!The domain of a function is all the valid inputs, or xxx values, we can plug in.Since we're dealing with a logarithm, the argument *inside* the logarithm must be greater than zero.In this case, the argument is x+6x+6x+6.

STEP 4

So, we need x+6>0x+6 > 0x+6>0.To **isolate** xxx, we'll subtract 6 from both sides of the inequality: x+6−6>0−6x + 6 - 6 > 0 - 6x+6−6>0−6.This simplifies to x>−6x > -6x>−6.Boom! That's our **domain**: x>−6x > -6x>−6.

STEP 5

Now, let's **tackle the range**!The range is all the possible outputs, or yyy values, our function can produce.Think about the graph of y=log⁡(x)y = \log(x)y=log(x).It extends infinitely in both the positive and negative yyy directions!

STEP 6

Our function is f(x)=log⁡(x+6)−4f(x) = \log(x+6) - 4f(x)=log(x+6)−4.The −4-4−4 just shifts the graph *down* by **4 units**.Shifting the graph up or down doesn't change the fact that the logarithm can still reach any yyy value.

STEP 7

Therefore, the **range** of our function is all real numbers.In fancy math terms, we write that as −∞<y<∞-\infty < y < \infty−∞<y<∞.

STEP 8

The **domain** of f(x)=log⁡(x+6)−4f(x) = \log(x+6) - 4f(x)=log(x+6)−4 is x>−6x > -6x>−6, and the **range** is all real numbers.

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QuestionWhat are the domain and range of $f(x)=\log (x+6)-4$ ? domain: $x>-6$ ; range: $y>4$ domain: $x>-6$ ; range: all real numbers domain: $x>6$ ; range: $y>-4$ domain: $x>6$ ; range: all real numbers

What is this asking? We need to find the allowed $x$ values (domain) and the possible $y$ values (range) of the function $f(x) = \log(x+6) - 4$ . 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 valid inputs, or $x$ values, we can plug in.
Since we're dealing with a logarithm, the argument inside the logarithm must be greater than zero.
In this case, the argument is $x+6$ .

So, we need $x+6 > 0$ .
To isolate $x$ , we'll subtract 6 from both sides of the inequality: $x + 6 - 6 > 0 - 6$ .
This simplifies to $x > -6$ .
Boom! That's our domain: $x > -6$ .

Now, let's tackle the range!
The range is all the possible outputs, or $y$ values, our function can produce.
Think about the graph of $y = \log(x)$ .
It extends infinitely in both the positive and negative $y$ directions!

Our function is $f(x) = \log(x+6) - 4$ .
The $-4$ just shifts the graph down by 4 units.
Shifting the graph up or down doesn't change the fact that the logarithm can still reach any $y$ value.

Therefore, the range of our function is all real numbers.
In fancy math terms, we write that as $-\infty < y < \infty$ .

The domain of $f(x) = \log(x+6) - 4$ is $x > -6$ , and the range is all real numbers.