Question

perform the indicated operations in each exercise.

1. if (f(x) = 3x^2 - x + 5), find (f(0), f(3), f(-5), f(-3x)) and (f(x+1)).
2. if (f(x) = 9 + x^2), find (f(-2), f(2), f(-y), f(4y)) and (f(y+1)).
3. if (g(x) = \frac{x^3+3}{x+1}), find (g(1), g(-1), g(-y), g(2y)) and (g(0)).
4. if (v(y) = sin y), find (v(0), v(pi), v(pi-x), v(-x)) and (v(2pi)).
5. if (f(y) = sin y + cos y), find (f(pi), f(0), f(2pi), f(-y)) and (f(pi+x)).
6. if (v(x) = x(x^2+5)), find (v(0), v(-2), vleft(\frac{1}{2}

ight), v(-5)) and (v(y+4)).

1. if (g(y) = \tan y), find (g(pi), gleft(\frac{pi}{3}

ight), g(2x), g(-x)) and (gleft(\frac{pi}{6}
ight)).

1. if (f(\theta) = cos\theta), find (f(0), fleft(\frac{pi}{4}

ight), fleft(\frac{pi}{2}
ight), f(pi)) and (f(2pi)).

1. if (f(x) = (sec^2 x - 1)), find (f(0), fleft(\frac{pi}{4}

ight), fleft(\frac{pi}{3}
ight), f(pi)) and (f(2pi)).

1. if (f(x) = ln x), find (f(1), f(e), f(e^2), f(e^{-x})) and (f(cos 2pi)).
2. if (f(x) = e^x), find (f(0), f(1), f(-1), f(ln x)) and (f(lnsin x)).
3. if (g(y) = 1 + cos y), find (g(0), gleft(\frac{pi}{2}

ight), g(pi), g(2pi)) and (g(x+1)).

1. if (f(x) = \frac{x^2+1}{x}), find (f(0), f(e^x), f(\tan x), f(sin x + cos x)) and (f(sin^2 x)).
2. if (g(x) = \frac{(x+1)^2}{x}), find (gleft(\frac{1}{2}

ight), gleft(\frac{a}{b}
ight), g(sin x), g(\tan x)) and (g(e^x)).

1. if (g(y) = 2y + sin y), find (g(0), gleft(\frac{pi}{2}

ight), g(pi), g(2pi)) and (g(pi+x)).

1. express the area of a triangle as a function of its altitude having a base of 4 units. graph the function.
2. express the volume of a right circular cone with an altitude of 6 units as a function of its radius. graph the function.
3. express the hypotenuse of a right triangle as a function of its adjacent side having a 2.5 cm opposite side. graph the function.
4. express the area of an ellipse having a length of its major axis 9 units as a function of the length of the minor axis. graph the function.
5. express the height of a right circular cylinder as a function of its volume having a radius of 4 cm. graph the function.
6. express the radius of a sphere as a function of its volume. graph the function.
7. express the radius of a sphere as a function of its surface area. graph the function.
8. a right circular cylinder has an altitude of 10 m. express the radius of the cylinder as a function of its volume. graph the function.
9. a right circular cone uses 20 in(^2) of metal sheet. find the volume as a function of its radius. graph the function.
10. two sides of a triangle are 15 cm and 18 cm respectively. express the angle between the two sides as a function of its area. graph the function.
11. a parabola has an altitude of 4 units. express the length of its base as a function of its area. graph the function.
12. express the volume of a cube as a function of its surface area having a length of its side 5 cm. graph the function.
13. an ellipse has a 6 units length minor axis. express the volume of an ellipsoid generated by the ellipse as a function of the area of the ellipse. graph the function.
14. a population of a certain place is given by (p = 8000e^{0.457t}), where (p) is the present population and (t) is the time in years. express the time (t) as a function of the population (p). graph the function.
15. at a distance of 100 km, express the speed (v) as a function of time (t). graph the function.

Explanation:

⚡ Using what you learned: representing functions (tables, graphs, equations, mappings)

Step 1: Evaluate \( f(0) \)

Substitute \( x = 0 \) into the function \( f(x) = 3x^2 - x + 5 \):

$$ f(0) = 3(0)^2 - (0) + 5 = 5 $$

Step 2: Evaluate \( f(3) \)

Substitute \( x = 3 \) into the function:

$$ f(3) = 3(3)^2 - 3 + 5 = 3(9) - 3 + 5 = 27 - 3 + 5 = 29 $$

Step 3: Evaluate \( f(-5) \)

Substitute \( x = -5 \) into the function:

$$ f(-5) = 3(-5)^2 - (-5) + 5 = 3(25) + 5 + 5 = 75 + 10 = 85 $$

Step 4: Evaluate \( f(-3x) \)

Substitute the expression \( -3x \) in place of every \( x \) in the function:

$$ f(-3x) = 3(-3x)^2 - (-3x) + 5 = 3(9x^2) + 3x + 5 = 27x^2 + 3x + 5 $$

Step 5: Evaluate \( f(x + 1) \)

Substitute the expression \( x + 1 \) in place of every \( x \) in the function:

$$ f(x + 1) = 3(x + 1)^2 - (x + 1) + 5 $$

Expand the squared term and simplify:

$$ f(x + 1) = 3(x^2 + 2x + 1) - x - 1 + 5 $$

$$ f(x + 1) = 3x^2 + 6x + 3 - x + 4 $$

$$ f(x + 1) = 3x^2 + 5x + 7 $$

Answer:

For Exercise 11, where \( f(x) = 3x^2 - x + 5 \):

• \( f(0) = 5 \)
• \( f(3) = 29 \)
• \( f(-5) = 85 \)
• \( f(-3x) = 27x^2 + 3x + 5 \)
• \( f(x + 1) = 3x^2 + 5x + 7 \)