Problem 36 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

\(f(x)=5^x+4\)

Step-by-Step Solution

Answer

The solution to the function \(f(x) = 5^x + 4\) is \(f(x) = 5^x + 4\), which describes the relationship between \(x\) and \(f(x)\).

Step 1: Identify the unknown

Since the value of \(x\) is not provided, it will be represented as a general variable.

Step 2: Understand the function

The function \(f(x) = 5^x + 4\) means that for a given \(x\), the output or \(f(x)\) is obtained by exponentiating 5 by \(x\) and adding 4 to the result.

Step 3: Apply exponent rules

So, for any given value of \(x\), the operation will be to raise 5 to the power of \(x\) and then add 4 to the result.

Step 4: Calculate \(f(x)\)

As \(x\) is not specified, a specified value for \(x\) can't be calculated. Therefore, the result is a function that describes the relationship between \(x\) and \(f(x)\), expressed as \(f(x) = 5^x + 4\)

Key Concepts

Exponential Growth

Understanding exponential growth begins with recognizing its distinct behavior in various contexts, such as populations, investments, and even in mathematical functions. The exponential function, represented by the formula f(x) = a^x, where a is a positive constant, shows how quantities increase rapidly over time or as the variable x changes.

An essential characteristic of exponential growth is that the rate of change is proportional to the current amount. This means as the value of x increases, the output of f(x) rises exponentially, not linearly. In the given exercise, the base of the exponential term is 5, signifying that for each unit increase in x, the value of 5^x multiplies by five. The addition of 4 does not affect the exponential growth rate but simply shifts the entire function up by four units on the graph.

To visualize exponential growth, imagine a graph where the x-axis represents time or our variable x, and the y-axis shows the growth of the function f(x). The curve will start relatively flat at lower values of x, but as x becomes larger, the curve steepens dramatically, illustrating the explosive increase typical of exponential functions.

Function Evaluation

Function evaluation is the process of determining the output of a function for a particular input value. It is one of the fundamental concepts in mathematics, essential for understanding how functions work.

In the context of the exponential function f(x) = 5^x + 4, to evaluate the function, we follow a simple sequence of steps. First, we substitute the given value of x into the function. Next, we calculate the exponentiation 5^x, and finally, we add 4 to the result. The result we get after this process is the value of f(x).

Without a specified value for x, the evaluation is abstract. However, should a value be given, the calculation becomes concrete. For example, if x = 2, then f(2) = 5^2 + 4 = 25 + 4 = 29. Through function evaluation, we can graph this function or use it to model real-world scenarios like compound interest, population growth, or the spread of a virus, where the variable x often represents time.

Exponent Rules

Exponent rules are mathematical guidelines that make working with exponential expressions simpler and are pivotal in solving various problems involving exponents. There are several key rules which include the product rule, quotient rule, power rule, zero exponent rule, and negative exponent rule. An understanding of these rules assists students in correctly manipulating and simplifying exponential expressions.

In the given exercise f(x) = 5^x + 4, applying these exponent rules correctly is essential for simplifying any computations involving the function. For example, when calculating the function for x values, we use the understanding that raising a number to the power of zero gives us 1 (5^0 = 1). Also, when dealing with multiple exponential terms with the same base, as in 5^x * 5^y, we can add the exponents.

By mastering exponent rules, students can evaluate exponential functions more efficiently and can also handle more complex operations involving multiple terms with exponents. For instance, if one were to expand the function f(x) to include additional exponential terms, say f(x) = 5^x + 4 - 5^(x-1), the exponent rules become critical in simplifying this expression to find a more manageable form.