Math Problem Statement

Evaluate the following: a) Given a(x) = x^2 + 2x - 5, find a(x+3); b) Given b(x) = 5x + 3, find b(x^2 - 4)

Solution

Let's go through the two problems step by step.

Problem (a):

We are given a(x)=x2+2x5a(x) = x^2 + 2x - 5 and asked to find a(x+3)a(x+3).

To do this, we substitute x+3x+3 into the expression for a(x)a(x).

a(x+3)=(x+3)2+2(x+3)5a(x+3) = (x+3)^2 + 2(x+3) - 5

Now, expand and simplify:

(x+3)2=x2+6x+9(x+3)^2 = x^2 + 6x + 9 2(x+3)=2x+62(x+3) = 2x + 6 Substitute these back into the expression:

a(x+3)=(x2+6x+9)+(2x+6)5a(x+3) = (x^2 + 6x + 9) + (2x + 6) - 5

Simplify:

a(x+3)=x2+6x+9+2x+65a(x+3) = x^2 + 6x + 9 + 2x + 6 - 5 a(x+3)=x2+8x+10a(x+3) = x^2 + 8x + 10

Problem (b):

We are given b(x)=5x+3b(x) = 5x + 3 and asked to find b(x24)b(x^2 - 4).

To solve, we substitute x24x^2 - 4 into the expression for b(x)b(x).

b(x24)=5(x24)+3b(x^2 - 4) = 5(x^2 - 4) + 3

Now, expand:

b(x24)=5x220+3b(x^2 - 4) = 5x^2 - 20 + 3 b(x24)=5x217b(x^2 - 4) = 5x^2 - 17

Final Answers:

  • (a) a(x+3)=x2+8x+10a(x+3) = x^2 + 8x + 10
  • (b) b(x24)=5x217b(x^2 - 4) = 5x^2 - 17

Would you like more details on these solutions or have any questions? Here are some related questions you could explore:

  1. What would be the result if a(x)=3x2+4x2a(x) = 3x^2 + 4x - 2 instead?
  2. How would you solve a(2x+1)a(2x+1) for the same function a(x)a(x)?
  3. What is the general method for evaluating a(f(x))a(f(x)) for any function f(x)f(x)?
  4. How can this process be applied to more complex functions, such as logarithmic or trigonometric functions?
  5. If b(x)b(x) were a quadratic function, how would you approach b(x24)b(x^2 - 4)?

Tip: When substituting into a function, always expand carefully and combine like terms to avoid errors.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Function Substitution
Quadratics

Formulas

a(x) = x^2 + 2x - 5
b(x) = 5x + 3

Theorems

Substitution in Algebraic Expressions

Suitable Grade Level

Grades 9-10