Given the function
f
(
x
)
=
1
2
x
−
3
and the function
g
(
x
)
=
7
x
2
+
4
x
+
5
determine each of the following.

Give your answer as an integer or a simplified fraction.
Evaluate
f
(
g
(
2
)
)

f
(
g
(
2
)
)
=

Evaluate
g
(
f
(
4
)
)

g
(
f
(
4
)
)
=

Evaluate
f
(
g
(
1
4
)
)

f
(
g
(
1
4
)
)
=

Evaluate
f
(
f
(
1
2
)
)

f
(
f
(
1
2
)
)
=

Here are the evaluated results:

1. \( f(g(2)) = 17.5 \)
2. \( g(f(4)) = 8 \)
3. \( f(g\left(\frac{1}{4}\right)) = 0.21875 \)
4. \( f(f\left(\frac{1}{2}\right)) = -4.375 \)

Would you like more details or have any further questions?

Here are five related questions to consider:
1. How would the result change if we evaluate \( g(f(g(x))) \)?
2. Can we find \( f(g(-1)) \) based on these functions?
3. What is the derivative of \( f(x) \) and \( g(x) \)?
4. Can you determine the x-values for which \( f(g(x)) = 0 \)?
5. What are the points of intersection between \( f(x) \) and \( g(x) \)?

**Tip:** When dealing with nested functions, it's helpful to evaluate them step by step, beginning with the innermost function.

Given the function f(x) = 1/2x - 3 and the function g(x) = 7x^2 + 4x + 5, determine each of the following. Evaluate f(g(2)), g(f(4)), f(g(1/4)), and f(f(1/2)).

Explore how to evaluate function compositions involving linear and quadratic functions with f(x) = 1/2x - 3 and g(x) = 7x^2 + 4x + 5. This includes examples of f(g(2)), g(f(4)), and more. Suitable for students in Grades 9-12.

Math Problem Statement

Given the function f ( x )

1 2 x − 3 and the function g ( x )

f ( g ( 2 ) )

g ( f ( 4 ) )

f ( g ( 1 4 ) )

f ( f ( 1 2 ) )

Solution

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