Problem 8 - Calculus

Question

For the functions \(f\) and \(g\) find (a) \(f(g(1))\) (b) \(g(f(1))\) (c) \(f(g(x))\) (d) \(g(f(x))\) (e) \(f(t) g(t)\) $$f(x)=x^{2}, g(x)=x+1$$

Step-by-Step Solution

Answer

(a) 4, (b) 2, (c) x^2 + 2x + 1, (d) x^2 + 1, (e) t^3 + t^2

Step 1: Evaluate g(1)

To find \(f(g(1))\), we first need to calculate \(g(1)\) using the definition of the function \(g(x) = x + 1\). Substitute \(x = 1\): \[ g(1) = 1 + 1 = 2 \]

Step 2: Evaluate f(g(1))

Now, substitute \(g(1) = 2\) into the function \(f(x) = x^2\). \[ f(g(1)) = f(2) = 2^2 = 4 \]

Step 3: Evaluate f(1)

To find \(g(f(1))\), we first need to calculate \(f(1)\) using the definition of the function \(f(x) = x^2\). Substitute \(x = 1\): \[ f(1) = 1^2 = 1 \]

Step 4: Evaluate g(f(1))

Now, substitute \(f(1) = 1\) into the function \(g(x) = x + 1\). \[ g(f(1)) = g(1) = 1 + 1 = 2 \]

Step 5: Expression for f(g(x))

Substitute \(g(x) = x + 1\) into \(f(x) = x^2\) to find \(f(g(x))\): \[ f(g(x)) = f(x + 1) = (x + 1)^2 = x^2 + 2x + 1 \]

Step 6: Expression for g(f(x))

Substitute \(f(x) = x^2\) into \(g(x) = x + 1\) to find \(g(f(x))\): \[ g(f(x)) = g(x^2) = x^2 + 1 \]

Step 7: Expression for f(t)g(t)

The expression \(f(t)g(t)\) is obtained by multiplying the functions \(f(t) = t^2\) and \(g(t) = t + 1\): \[ f(t)g(t) = t^2(t + 1) = t^3 + t^2 \]

Key Concepts

Function Evaluation

Evaluating a function is the process of finding the output value of a function for a given input. In simpler terms, it's about determining what result you get when you plug a specific number into the function.When working with function evaluation, always follow these steps:

• First, identify the function you are dealing with, such as \( f(x) = x^2 \) or \( g(x) = x + 1 \).
• Next, substitute the given input value into the function. This will help you to find the output.

For example, in the case of \( f(g(1)) \), we first evaluate \( g(1) \). Doing this provides us with a value to substitute into \( f(x) \). Evaluating \( f(2) \) results in the final answer of 4. Breaking it down into separate evaluations, makes it straightforward and prevents errors.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients combined through addition, subtraction, and multiplication. They are characterized by terms with non-negative integer exponents.Polynomials can be categorized based on their degree, which is the highest exponent present in the expression.

• A polynomial of degree 2, like \( x^2 \), is called a quadratic polynomial.
• A polynomial like \( x^3 \) is a cubic polynomial.

In the exercise, we used the polynomial \( f(x) = x^2 \). When calculating \( f(g(x)) = (x + 1)^2 \), the original expression expands to \( x^2 + 2x + 1 \). This is a result of applying the distributive property, a common technique in algebra.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the basis of most algebra problems, including those involving functions and polynomials.When working with algebraic expressions, it's essential to know how to manipulate them efficiently:

• To combine like terms, add or subtract terms that have the same variable raised to the same power.
• To factor expressions, rewrite them as a product of simpler expressions.

In the exercise, combining \( f(t)g(t) \) into a single expression involves multiplication. The expressions \( t^2 \) and \( t + 1 \) are multiplied to create one comprehensive polynomial: \( t^3 + t^2 \). Each part of the expression interacts with others according to the rules of algebraic expressions. Understanding these foundational rules makes it easier to handle more complex algebraic problems.