Problem 17 - Precalculus

Question

For the functions \(f(x)=3^{x}, g(x)=\left(\frac{1}{16}\right)^{x},\) and \(h(x)=10^{x+1},\) find the function value at the indicated points. $$g\left(-\frac{1}{2}\right)$$

Step-by-Step Solution

Answer

The function value is 4.

Step 1: Identify the function

We are given the function to evaluate: \( g(x) = \left(\frac{1}{16}\right)^x \). This is an exponential function with base \( \frac{1}{16} \).

Step 2: Substitute the value into the function

Let's substitute \( x = -\frac{1}{2} \) into the function: \[ g\left(-\frac{1}{2}\right) = \left( \frac{1}{16} \right)^{-\frac{1}{2}} \].

Step 3: Rewrite the expression using positive exponents

We can convert the negative exponent by taking the reciprocal of the base: \[ \left( \frac{1}{16} \right)^{-\frac{1}{2}} = \left( 16 \right)^{\frac{1}{2}} \].

Step 4: Evaluate the expression

The expression \( 16^{\frac{1}{2}} \) means the square root of 16. Since \( \sqrt{16} = 4 \), we find that the value is \( 4 \).

Step 5: Conclusion

The function value of \( g \left( -\frac{1}{2} \right) \) is \( 4 \).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics. When we evaluate a function, we substitute a specific value in place of the variable. In the context of exponential functions, like the given function \( g(x) = \left(\frac{1}{16}\right)^x \), the process involves substituting the given \( x \) value into the function and simplifying.

• Identify the function and the evaluation point.
• Substitute the \( x \) value into the function.
• Simplify the expression to get the result.For our specific example, we substitute \(-\frac{1}{2}\) into the function \( g(x) \). This results in \( g\left(-\frac{1}{2}\right) = \left( \frac{1}{16} \right)^{-\frac{1}{2}} \).

The resulting expression can then be simplified by employing the concepts of negative exponents and square roots.

Negative Exponents

Negative exponents can seem tricky at first, but they follow a simple rule: a negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. For example, a function with a negative exponent like \( \left(\frac{1}{16}\right)^{-\frac{1}{2}} \) is equivalent to \( \left(16\right)^{\frac{1}{2}} \).

Steps to Simplify Negative Exponents:

• Identify the base and the negative exponent.
• Take the reciprocal of the base. The reciprocal of a number \( a \) is \( \frac{1}{a} \).
• Change the sign of the exponent to positive.

In this case, the reciprocal of \( \frac{1}{16} \) is 16, which gives us \( 16^{\frac{1}{2}} \) as the equivalent expression with a positive exponent.

Square Roots

Square roots are a special mathematical operation that find a number which, when multiplied by itself, equals the original number. When looking at a function like \( 16^{\frac{1}{2}} \), it is expressing the square root: \( \sqrt{16} \).

Understanding the Square Root:

• The square root of a number \( a \) is denoted as \( \sqrt{a} \).
• If \( a = b^2 \), then \( \sqrt{a} = b \).
• For our example, \( \sqrt{16} = 4 \) because \( 4 \times 4 = 16 \).

By understanding and applying the concept of square roots, we completed the evaluation of our function, yielding the final value of 4.