Question

1. if the point (k, 3) exists on the inverse of y = 2^x, the value of k is:

a. 2
b. 3
c. 4
d. 8

Explanation:

Step1: Recall inverse function property

If a point \((a, b)\) is on a function \(y = f(x)\), then \((b, a)\) is on its inverse function \(y = f^{-1}(x)\). So, if \((k, 3)\) is on the inverse of \(y = 2^{x}\), then \((3, k)\) is on \(y = 2^{x}\).

Step2: Substitute into original function

Substitute \(x = 3\) and \(y = k\) into \(y = 2^{x}\). We get \(k = 2^{3}\).

Step3: Calculate the value

Calculate \(2^{3}\), which is \(8\)? Wait, no, wait. Wait, maybe I made a mistake. Wait, the inverse of \(y = 2^{x}\) is \(y=\log_{2}x\) (since to find the inverse, we swap \(x\) and \(y\) and solve for \(y\): \(x = 2^{y}\) implies \(y=\log_{2}x\)). So if \((k, 3)\) is on the inverse function \(y = \log_{2}x\), then when \(y = 3\), \(x = k\). So substitute \(y = 3\) into \(y=\log_{2}x\), we get \(3=\log_{2}k\). By the definition of logarithms, this means \(k = 2^{3}=8\)? But the options have D as 8? Wait, but let's check again. Wait, the original function is \(y = 2^{x}\). The inverse function is \(x = 2^{y}\), so \(y=\log_{2}x\). So the point \((k, 3)\) is on the inverse, so \(3=\log_{2}k\), so \(k = 2^{3}=8\). But let's check the options. Option D is 8. Wait, but maybe I messed up the inverse property. Wait, another way: If \((k, 3)\) is on the inverse of \(y = 2^{x}\), then \((3, k)\) is on \(y = 2^{x}\). So substitute \(x = 3\) into \(y = 2^{x}\), we get \(y = 2^{3}=8\), so \(k = 8\). So the answer should be D. 8. Wait, but let's check the options again. The options are A.2, B.3, C.4, D.8. So yes, D is 8.

Wait, but maybe I made a mistake in the inverse property. Let's confirm: The inverse function of \(y = f(x)\) has the property that if \((a, b)\) is on \(f\), then \((b, a)\) is on \(f^{-1}\). So if \((k, 3)\) is on \(f^{-1}\), then \((3, k)\) is on \(f\). So \(f(3)=k\), and \(f(x)=2^{x}\), so \(f(3)=2^{3}=8\), so \(k = 8\). So the correct answer is D. 8.

Answer:

D. 8