Question

The function \(g(x)=16^x\) is defined for all real values of \(x\). What is \(g\left(\frac{1}{2}\right)+g\left(\frac{1}{4}\right) ?\) A) 2 B) 6 C) 8 D) 16

Short answer

The short answer to the problem is: \(g\left(\frac{1}{2}\right) + g\left(\frac{1}{4}\right) = 4 + 2 = 6\). So, the correct answer is B) 6.

Step-by-step solution

Find g(x) for \(x = \frac{1}{2}\)

To find g(x) for \(x = \frac{1}{2}\), substitute \(x = \frac{1}{2}\) in the given function: \[g\left(\frac{1}{2}\right) = 16^{\frac{1}{2}}\] Since \(16^{\frac{1}{2}}\) is the square root of 16, we have: \[g\left(\frac{1}{2}\right) = 4\]

Find g(x) for \(x = \frac{1}{4}\)

To find g(x) for \(x = \frac{1}{4}\), substitute \(x = \frac{1}{4}\) in the given function: \[g\left(\frac{1}{4}\right) = 16^{\frac{1}{4}}\] Since \(16^{\frac{1}{4}}\) is the fourth root of 16, we have: \[g\left(\frac{1}{4}\right) = 2\]

Add the values of g(x) found in steps 1 and 2

Now, we need to find the sum of the values of g(x) found in the previous steps: \[g\left(\frac{1}{2}\right) + g\left(\frac{1}{4}\right) = 4 + 2 = 6\]

Compare the sum with the answer choices

The sum of g(x) at the specific points is 6. Comparing it with the answer choices, we find that the correct answer is: B) 6

Key concepts

Understanding Exponential Functions

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In our exercise, the exponential function is represented as g(x) = 16^x. For any real value of x, the function calculates the power of 16 raised to that value.

In everyday life, exponential functions can describe rapid growth or decay, such as population growth, radioactive decay, or even compounding interest in finance. Understanding how to work with these functions allows us to interpret such real-world phenomena. When evaluating an exponential function at a fractional exponent like 1/2 or 1/4, the exponent indicates a root of the base. Specifically, g(1/2) corresponds to the square root of 16 and g(1/4) to the fourth root of 16.

Mastering Square Roots

A square root is essentially the opposite of squaring a number. If you square 2, you get 4. Conversely, the square root of 4 is 2. This operation is crucial to know, as square roots often arise in algebra, geometry, and various applied sciences.

In the step-by-step solution for our problem, we evaluate g(1/2) to find the square root of 16, which is 4. It's equally important to know that every positive number has two square roots: one positive and the other negative. However, in the context of our SAT Math problem, the negative root is not considered since we are dealing with function outputs that represent real-world quantities, which in many cases cannot be negative.

Fourth Roots Simplified

Similar to square roots, fourth roots are part of the family of nth roots. The fourth root of a number is a value that, when raised to the power of four, gives the original number. For instance, the fourth root of 16 is 2, because 2^4 = 16.

Understanding fourth roots is slightly less intuitive than square roots, but they appear in various mathematical contexts, including geometry and calculus. It's useful to remember that perfect squares like 16, have very straightforward fourth roots since they are the square of whole numbers. In our case, since 4 times itself gives 16, and 2 times itself gives 4, 2 is the fourth root of 16 that we use in our problem to calculate g(1/4). This knowledge can be especially helpful when dealing with radical expressions and simplifying algebraic problems on the SAT.