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.

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

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

• \\(f(x)\\) approaches negative infinity

what happens to the value of \\(f(x) = \log_4 x\\) as \\(x\\) approaches \\(+\infty\\)?

Explanation:

Analyze the y-intercept of the logarithmic function

Using the Logarithmic Intercepts and Logarithmic and Exponential Inverses knowledge points
To find the \(y\)-intercept of \(f(x) = \log_4 x\), we evaluate the function at \(x = 0\):

$$ f(0) = \log_4(0) $$

This requires finding a power \(y\) such that \(4^y = 0\). Since exponential functions of the form \(y = 4^x\) are always strictly positive, there is no real power of \(4\) that equals \(0\).

Furthermore, the inverse of \(f(x) = \log_4 x\) is \(f^{-1}(x) = 4^x\). A \(y\)-intercept on \(f(x)\) corresponds to an \(x\)-intercept on its inverse \(f^{-1}(x)\). Since \(4^x = 0\) has no solution, the inverse function \(f^{-1}(x) = 4^x\) has no \(x\)-intercepts.

Therefore, the correct explanations are:

• There is no power of 4 that is equal to 0.
• Its inverse does not have any x-intercepts.

Determine the x-intercept of the logarithmic function

Using the Logarithmic Intercepts knowledge point
To find the \(x\)-intercept, we set \(f(x) = 0\):

$$ \log_4 x = 0 \implies x = 4^0 = 1 $$

Thus, the \(x\)-intercept of \(f(x) = \log_4 x\) is \(1\).

Analyze the limit behavior as x approaches 0 from the right

Using the Logarithmic Limits and Asymptotic Behavior of Logarithms knowledge points
As \(x\) approaches \(0\) from the positive side (\(x \to 0^+\)), the power to which \(4\) must be raised to yield \(x\) becomes increasingly negative:

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

Thus, \(f(x)\) approaches negative infinity.

Analyze the limit behavior as x approaches positive infinity

Using the Logarithmic Limits and Asymptotic Behavior of Logarithms knowledge points
As \(x\) increases without bound (\(x \to +\infty\)), the value of the logarithm also increases without bound, albeit slowly:

$$ \lim_{x \to +\infty} \log_4 x = +\infty $$

Thus, \(f(x)\) approaches positive infinity.

Answer:

Question 1

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.
• Its inverse does not have any x-intercepts.

Question 2

The \(x\)-intercept of \(f(x) = \log_4 x\) is 1.

Question 3

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

• \(f(x)\) approaches negative infinity

Question 4

What happens to the value of \(f(x) = \log_4 x\) as \(x\) approaches \(+\infty\)?

• \(f(x)\) approaches positive infinity