Math Problem Statement

Evaluate the following functions and show your solutions: 1) g(x) = 5x - 7; g(x+1), 2) h(x) = sqrt(x^2 + 2x + 4); h(2), 3) k(x) = (3x-2)/(2x+4); k(-3), 4) f(x) = 2x^2 + 5x - 9; f(5x-2), 5) g(p) = 4^p; g(p * 3/2).

Solution

I see the image contains several mathematical problems. Let me break down and solve each one.

1. g(x)=5x7g(x) = 5x - 7

  • We are asked to evaluate g(x+1)g(x+1).
  • Substitute x+1x+1 into the function g(x)g(x): g(x+1)=5(x+1)7=5x+57=5x2g(x+1) = 5(x+1) - 7 = 5x + 5 - 7 = 5x - 2
  • Answer: g(x+1)=5x2g(x+1) = 5x - 2.

2. h(x)=x2+2x+4h(x) = \sqrt{x^2 + 2x + 4}

  • Evaluate h(2)h(2): h(2)=22+2(2)+4=4+4+4=12=23h(2) = \sqrt{2^2 + 2(2) + 4} = \sqrt{4 + 4 + 4} = \sqrt{12} = 2\sqrt{3}
  • Answer: h(2)=23h(2) = 2\sqrt{3}.

3. k(x)=3x22x+4k(x) = \frac{3x - 2}{2x + 4}

  • Evaluate k(3)k(-3): k(3)=3(3)22(3)+4=926+4=112=112k(-3) = \frac{3(-3) - 2}{2(-3) + 4} = \frac{-9 - 2}{-6 + 4} = \frac{-11}{-2} = \frac{11}{2}
  • Answer: k(3)=112k(-3) = \frac{11}{2}.

4. f(x)=2x2+5x9f(x) = 2x^2 + 5x - 9

  • Evaluate f(5x2)f(5x - 2): f(5x2)=2(5x2)2+5(5x2)9f(5x - 2) = 2(5x - 2)^2 + 5(5x - 2) - 9 First, expand (5x2)2(5x - 2)^2: (5x2)2=25x220x+4(5x - 2)^2 = 25x^2 - 20x + 4 Now substitute and simplify: f(5x2)=2(25x220x+4)+5(5x2)9f(5x - 2) = 2(25x^2 - 20x + 4) + 5(5x - 2) - 9 f(5x2)=50x240x+8+25x109f(5x - 2) = 50x^2 - 40x + 8 + 25x - 10 - 9 f(5x2)=50x215x11f(5x - 2) = 50x^2 - 15x - 11
  • Answer: f(5x2)=50x215x11f(5x - 2) = 50x^2 - 15x - 11.

5. g(p)=4pg(p) = 4^p

  • Evaluate g(p32)g\left( p \cdot \frac{3}{2} \right). g(p32)=4p32=(4p)32=43p/2g\left( p \cdot \frac{3}{2} \right) = 4^{p \cdot \frac{3}{2}} = (4^p)^{\frac{3}{2}} = 4^{3p/2}
  • Answer: g(p32)=43p/2g\left( p \cdot \frac{3}{2} \right) = 4^{3p/2}.

Would you like further details on any specific problem?

Here are some related questions for practice:

  1. Can you generalize the evaluation for any function g(x+c)g(x+c) given g(x)=ax+bg(x) = ax + b?
  2. What is the domain and range of h(x)=x2+2x+4h(x) = \sqrt{x^2 + 2x + 4}?
  3. For the function k(x)k(x), what values of xx would make the function undefined?
  4. Expand and simplify f(3x1)f(3x - 1) if f(x)=2x2+5x9f(x) = 2x^2 + 5x - 9.
  5. What happens to g(p)g(p) when p=0p = 0?

Tip:

When simplifying functions, always expand carefully and check for factorizations that might make calculations simpler.

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Math Problem Analysis

Mathematical Concepts

Function Evaluation
Quadratic Expressions
Square Roots
Rational Functions
Exponential Functions

Formulas

g(x) = ax + b
h(x) = sqrt(ax^2 + bx + c)
k(x) = (ax + b)/(cx + d)
f(x) = ax^2 + bx + c
g(p) = a^p

Theorems

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Suitable Grade Level

Grades 10-12