Questions: Evaluate each of the following expressions given the function definitions for (f, g), and (h).
(f(x)=sqrtx+4)
(g(x)=4x-8)
(h(x)=frac7x)
a. (f(g(4))=)
b. (g(f(72))=)
c. (h(g(f(2)))=)
Transcript text: Evaluate each of the following expressions given the function definitions for $f, g$, and $h$.
\[
\begin{array}{l}
f(x)=\sqrt{x+4} \\
g(x)=4 x-8 \\
h(x)=\frac{7}{x}
\end{array}
\]
a. $f(g(4))=$ $\square$ Preview
b. $g(f(72))=$ $\square$ Preview
c. $h(g(f(2)))=$ $\square$
Preview
Solution
Solution Steps
To solve these problems, we need to evaluate the given function compositions step by step. For each part, we will substitute the inner function's result into the outer function.
a. Evaluate f(g(4)): First, calculate g(4) and then use this result to find f(g(4)).
b. Evaluate g(f(72)): First, calculate f(72) and then use this result to find g(f(72)).
c. Evaluate h(g(f(2))): First, calculate f(2), then use this result to find g(f(2)), and finally use this result to find h(g(f(2))).
Step 1: Evaluate g(4)
To find f(g(4)), we first evaluate g(4). Using the function definition g(x)=4x−8, we have:
g(4)=4×4−8=16−8=8
Step 2: Evaluate f(g(4))
Next, we substitute g(4)=8 into the function f(x)=x+4:
f(g(4))=f(8)=8+4=12≈3.464
Step 3: Evaluate f(72)
To find g(f(72)), we first evaluate f(72). Using the function definition f(x)=x+4, we have:
f(72)=72+4=76≈8.718
Step 4: Evaluate g(f(72))
Next, we substitute f(72)≈8.718 into the function g(x)=4x−8:
g(f(72))=g(8.718)=4×8.718−8=34.872−8=26.872
Step 5: Evaluate f(2)
To find h(g(f(2))), we first evaluate f(2). Using the function definition f(x)=x+4, we have:
f(2)=2+4=6≈2.449
Step 6: Evaluate g(f(2))
Next, we substitute f(2)≈2.449 into the function g(x)=4x−8:
g(f(2))=g(2.449)=4×2.449−8=9.796−8=1.798
Step 7: Evaluate h(g(f(2)))
Finally, we substitute g(f(2))≈1.798 into the function h(x)=x7:
h(g(f(2)))=h(1.798)=1.7987≈3.893