Problem 5 - Calculus with Concepts in Calculus

Question

Let \(f(x)=2 x^{2}+x-4\) and \(g(x)=3-x^{2} .\) Find the specified values. $$ f(g(1)) $$

Step-by-Step Solution

Answer

The value of \( f(g(1)) \) is 6.

Step 1: Evaluate g(1)

First, we need to find the value of the function \( g(x) \) at \( x = 1 \). Substitute \( x = 1 \) into the function \( g(x) = 3-x^2 \).\[g(1) = 3 - (1)^2 = 3 - 1 = 2\]So, \( g(1) = 2 \).

Step 2: Evaluate f(g(1))

Now that we have \( g(1) = 2 \), we substitute \( g(1) \) into \( f(x) \). This means we need to evaluate \( f(2) \). The function \( f(x) = 2x^2 + x - 4 \) will be used.\[f(2) = 2(2)^2 + 2 - 4\]Calculate each part:- \( 2(2)^2 = 2 \times 4 = 8 \)- \( 8 + 2 = 10 \)- \( 10 - 4 = 6 \)So, \( f(2) = 6 \).

Key Concepts

Quadratic Functions

Quadratic functions are a key concept in algebra that describe a special type of relationship between variables. You can recognize a quadratic function by its standard form: \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). This form creates a parabolic graph that can open either upwards or downwards, depending on the sign of \( a \).
In the context of our exercise, both \( f(x) = 2x^2 + x - 4 \) and \( g(x) = 3-x^2 \) are quadratic functions. Here, \( f(x) \) has coefficients \( a = 2 \), \( b = 1 \), and \( c = -4 \). The parabola opens upwards because \( a = 2 \) is positive. Conversely, \( g(x) \) has \( a = -1 \), indicating the parabola opens downwards due to the negative sign of \( a \).
Quadratics can model various real-world situations, from projectile motion to optimization problems.

Evaluating Functions

Evaluating a function means finding its output for a given input. Simply substitute the input value into the expression and solve for the result.
For instance, to evaluate \( g(1) \), you substitute \( x = 1 \) into \( g(x) = 3 - x^2 \):
\[ g(1) = 3 - (1)^2 = 3 - 1 = 2 \]
Thus, \( g(1) = 2 \).
Similarly, after finding \( g(1) = 2 \), you must evaluate \( f(2) \). Substitute \( x = 2 \) into \( f(x) = 2x^2 + x - 4 \):
\[ f(2) = 2(2)^2 + 2 - 4 \]
Simplifying, you calculate \( 2(2)^2 = 8 \), then add 2, resulting in 10, and finally subtract 4 to get 6. So, \( f(2) = 6 \).

• This step-by-step calculation helps ensure accuracy.
• Always perform the operations in parentheses and follow the order of operations: exponentiation, multiplication and division, addition, and subtraction.

Substitution

Substitution is a vital algebraic method utilized in evaluating composite functions, among other things. It involves replacing variables with corresponding values or expressions.
In our exercise, we use substitution to solve \( f(g(1)) \):

• First, determine \( g(1) \) by substituting \( x = 1 \) into the function \( g(x) \), which results in \( g(1) = 2 \).
• Then, substitute \( g(1) = 2 \) back into the function \( f(x) \). Hence, evaluate \( f(2) \) using the expression \( f(x) = 2x^2 + x - 4 \).

This process is a straightforward approach to handle function compositions like \( f(g(x)) \).
Substitution helps simplify complex expressions by breaking them down into more manageable steps. It also reinforces the idea of function chains, helping in understanding how functions can operate inside one another.