Problem 14 - 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. $$h(1)$$

Step-by-Step Solution

Answer

The value of \( h(1) \) is 100.

Step 1: Identify Function Definition

The function given is \( h(x) = 10^{x+1} \). We will use this function to find the value at \( x = 1 \).

Step 2: Substitute the Given Value

Substitute \( x = 1 \) into the function: \( h(1) = 10^{1+1} \).

Step 3: Simplify the Expression

Perform the arithmetic inside the exponent: \( 1 + 1 = 2 \). So, we now have \( h(1) = 10^{2} \).

Step 4: Compute the Power

Calculate \( 10^{2} \), which means \( 10 \times 10 = 100 \). Thus, \( h(1) = 100 \).

Key Concepts

Function Evaluation

Evaluating a function means finding the function's output for a given input. When working with functions, the input is typically represented by a variable such as \( x \). To evaluate a function at a specific point, you substitute the given value of \( x \) into the function and carry out any necessary calculations.
This allows you to determine the output value, often referred to as the function's value at that point.

• Consider the function \( h(x) = 10^{x+1} \). Here, the function is evaluated by replacing \( x \) with the value at which you wish to evaluate the function.
• For example, to find \( h(1) \), replace \( x \) with 1.

After substitution, simplify and compute the final result to find the output, which is the evaluation or value of the function.

Exponentiation

Exponentiation is a mathematical operation involving numbers called the base and the exponent. The exponent tells you how many times to multiply the base by itself. This operation is a key part of evaluating functions that include powers.
For example, in \( h(x) = 10^{x+1} \), the base is 10, and the exponent \( x+1 \) means you multiply 10 by itself \( x+1 \) times.

• When exponentiating, first simplify the expression involving the exponent. If the exponent is \( x+1 \), evaluate whatever \( x+1 \) equals first.
• Then calculate the power. For \( 10^2 \), the outcome is 100 because you multiply 10 by itself: \( 10 \times 10 = 100 \).

Exponentiation is crucial for simplifying calculations and solving problems involving exponential functions.

Function Substitution

Substitution in functions involves replacing a variable in the function's expression with a specific value. This is a straightforward but powerful technique to find how functions behave for different inputs.
Begin by identifying the variable and its current value within the function; then substitute the specific value into the equation.

• In \( h(x) = 10^{x+1} \), to find \( h(1) \), you replace every instance of \( x \) with 1: resulting in \( 10^{1+1} \).
• Always perform arithmetic operations inside parentheses or exponents first before continuing with further calculations.

Substitution helps to translate variables into numbers, giving you concrete values for analysis.