Which of the following attributes do the functions f(x)=√x and g(x)=log x have in common?
A) Both functions have a range of [0,∞).
B) Both functions have a domain of [0,∞).
C) Both functions have an x-intercept and a y-intercept.
D)Both functions are always increasing.
C) Both functions have an x-intercept and a y-intercept.
is that correct?
I apologize for the mistake. The correct answer is B) Both functions have a domain of [0,∞). Both functions only have real values for non-negative inputs.
To determine which attributes the functions \(f(x)=\sqrt{x}\) and \(g(x)=\log{x}\) have in common, let's evaluate each option:
A) Both functions have a range of \([0,\infty)\):
To find the range of a function, we need to determine the set of all possible output values. For \(f(x)=\sqrt{x}\), the square root of any non-negative number (including zero) will always be non-negative, so the range of \(f(x)\) is \([0,\infty)\). Similarly, for \(g(x)=\log{x}\), the logarithm of any positive number will always be real, so the range of \(g(x)\) is also \([0,\infty)\). Therefore, option A is correct.
B) Both functions have a domain of \([0,\infty)\):
To find the domain of a function, we need to determine the set of all possible input values. For \(f(x)=\sqrt{x}\), we can take the square root of any non-negative number (including zero), so the domain is \([0,\infty)\). However, for \(g(x)=\log{x}\), the logarithm is only defined for positive numbers, so the domain of \(g(x)\) is \((0,\infty)\). Therefore, option B is incorrect.
C) Both functions have an \(x\)-intercept and a \(y\)-intercept:
An \(x\)-intercept occurs when the function value is zero, which means \(f(x)=0\) or \(g(x)=0\). For \(f(x)=\sqrt{x}\), the only \(x\)-value that makes \(f(x)=0\) is \(x=0\), so it has an \(x\)-intercept. However, for \(g(x)=\log{x}\), there is no solution to the equation \(g(x)=0\) because the logarithm of zero is undefined. Therefore, option C is incorrect.
D) Both functions are always increasing:
To determine if a function is increasing, we need to check if the function values increase as the input values increase. For \(f(x)=\sqrt{x}\), as \(x\) increases, so does \(\sqrt{x}\), meaning the function is increasing. Similarly, for \(g(x)=\log{x}\), as \(x\) increases, the logarithm of \(x\) also increases. Therefore, both functions are always increasing. Therefore, option D is correct.
Therefore, the correct answer is:
A) Both functions have a range of \([0,\infty)\).
D) Both functions are always increasing.