This exercise is based on the following functions.

*f*(*x*) = *x*2 + 3 with domain (−∞, ∞)

*g*(*x*) = *x* − 3 with domain (−∞, ∞)

*h*(*x*) = *x* + 7 with domain [19, ∞)

*u*(*x*) = 

*x* + 19

 with domain [−19, 0)

*v*(*x*) = 

19 − *x*

 with domain [0, 19]

(a)

Write a formula for the indicated function.

*d* = *g* − *f*

*d*(*x*) = 

(x−3)−(x2+3)
 
  **Nice job!**

(b)

Give the domain. (Enter your answer using interval notation.)

(−∞,∞)
 
  **Nice job!**

(c)

Specify the function's value at the given point *a* = −1, if defined. (If an answer is undefined, enter UNDEFINED.)

*d*(−1) =  

Let's go through the steps to solve the problem:

### (a) Write a formula for the function \( d(x) = g(x) - f(x) \).

Given:
- \( f(x) = x^2 + 3 \)
- \( g(x) = x - 3 \)

The function \( d(x) \) is defined as:

\[
d(x) = g(x) - f(x)
\]

Substitute the given functions \( g(x) \) and \( f(x) \) into this formula:

\[
d(x) = (x - 3) - (x^2 + 3)
\]

Simplify the expression:

\[
d(x) = x - 3 - x^2 - 3 = -x^2 + x - 6
\]

**The formula for the function** \( d(x) \) **is** \( \mathbf{-x^2 + x - 6} \).

### (b) Give the domain of \( d(x) \).

Since \( d(x) \) is a polynomial function (a quadratic), it is defined for all real numbers. Therefore, the domain is:

\[
\text{Domain of } d(x) = (-\infty, \infty)
\]

### (c) Specify the function's value at the given point \( a = -1 \).

To find \( d(-1) \), substitute \( x = -1 \) into the formula for \( d(x) \):

\[
d(-1) = -(-1)^2 + (-1) - 6
\]

Calculate step-by-step:

\[
d(-1) = -(1) - 1 - 6 = -1 - 1 - 6 = -8
\]

**The value of** \( d(-1) \) **is** \( \mathbf{-8} \).

Would you like to explore any additional details or have any questions?

---

Here are 5 related questions you might find interesting:

1. How does the graph of \( d(x) \) compare to the graphs of \( f(x) \) and \( g(x) \)?
2. What are the roots (x-intercepts) of the function \( d(x) \)?
3. What is the vertex of the parabola represented by \( d(x) \)?
4. If \( g(x) \) were changed to \( g(x) = 2x - 3 \), how would that affect \( d(x) \)?
5. Can you determine the range of \( d(x) \)?

**Tip:** When combining functions like \( g(x) - f(x) \), always remember to distribute the negative sign across all terms of the second function to avoid mistakes.

Explore how to analyze and compute the function g(x) - f(x) using given functions f(x) = x^2 + 3 and g(x) = x - 3. Learn about function composition and domain specification with examples. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation