Question

which of the following explains why (f(x) = log_4 x) does not have a (y)-intercept? check all that apply.

there is no power of 4 that is equal to 0
there is no power of 4 that is equal to 1
its inverse does not have any (x)-intercepts
its inverse does not have any (y)-intercepts

what happens to the value of (f(x) = log_4 x) as (x) approaches 0 from the right?

(f(x)) approaches negative infinity
(f(x)) approaches 0
(f(x)) approaches infinity

the (x)-intercept of (f(x) = log_4 x) is 1

Explanation:

Analyze the y-intercept of the logarithmic function

Using the Logarithmic Functions and Logarithmic and Exponential Inverses knowledge points

$$
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$$

Determine the x-intercept of the logarithmic function

Using the Logarithmic Functions knowledge point

$$
LATEXBLOCK1
$$

Evaluate the limit behavior as x approaches 0 from the right

For the function \(f(x) = \log_4(x)\), as the input \(x\) approaches \(0\) from the positive side (\(x \to 0^+\)), the exponent \(y\) in \(4^y = x\) must become an increasingly large negative number to yield a value close to \(0\). Therefore:

$$ \lim_{x \to 0^+} \log_4(x) = -\infty $$

This means \(f(x)\) approaches negative infinity.

Answer:

Question 1

Which of the following explains why \(f(x) = \log_4x\) does not have a \(y\)-intercept? Check all that apply.

• There is no power of 4 that is equal to 0 (Correct answer)
• There is no power of 4 that is equal to 1
• Its inverse does not have any x-intercepts (Correct answer)
• Its inverse does not have any y-intercepts

Question 2

The \(x\)-intercept of \(f(x) = \log_4x\) is <blank>1</blank>.

Question 3

What happens to the value of \(f(x) = \log_4x\) as \(x\) approaches 0 from the right?

• f(x) approaches negative infinity (Correct answer)
• f(x) approaches 0
• f(x) approaches infinity