Math  /  Algebra

QuestionState the domain and the vertical asymptote of the function. f(x)=3log(x)+5f(x)=3 \log (-x)+5
Domain: \square (Enter your answer in interval notation.)
Vertical asymptote: x=x= \square Question Help: Written Example

Studdy Solution

STEP 1

What is this asking? We need to find the allowed xx values and the vertical line that the logarithmic function f(x)f(x) approaches but never touches! Watch out! Remember that we can only take the logarithm of *positive* numbers.
Also, don't mix up horizontal and vertical asymptotes!

STEP 2

1. Find the domain.
2. Find the vertical asymptote.

STEP 3

We know the argument of the logarithm *must* be positive, so we set up the inequality x>0-x > 0.
This is because we can only take the log of a positive number!

STEP 4

To solve for xx, we **multiply** both sides of the inequality by 1-1.
Remember that when we multiply or divide an inequality by a negative number, we must *flip* the inequality sign.
So, x>0-x > 0 becomes x<0x < 0.

STEP 5

This means xx can be any number less than zero.
In interval notation, this is written as (,0)(-\infty, 0).
This is our **domain**!

STEP 6

A vertical asymptote occurs when the function approaches infinity (or negative infinity) as xx approaches a specific value.
For logarithmic functions, this happens when the argument of the log approaches zero.

STEP 7

In our function f(x)=3log(x)+5f(x) = 3 \log (-x) + 5, the argument of the logarithm is x-x.
We set this equal to zero: x=0-x = 0.

STEP 8

Multiplying both sides by 1-1 gives us x=0x = 0.

STEP 9

Therefore, the vertical asymptote is x=0x = 0, which is the yy-axis!

STEP 10

Domain: (,0)(-\infty, 0) Vertical asymptote: x=0x = 0

Was this helpful?

Studdy solves anything!

banner

Start learning now

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

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord