Chapter 4

Write \(y\) as a function of \(x,\) where the value of \(y\) is equal to the given expression.The cube of \(x\)

Which of the following relations are also functions? Explain. $$y=3 x^{2}-5$$

Write \(y\) as a function of \(x,\) where the value of \(y\) is equal to the given expression.The square root of \(x\), diminished by 5.

Which of the following relations are also functions? Explain. $$y=\sqrt{2 x}$$

Write \(y\) as a function of \(x,\) where the value of \(y\) is equal to the given expression.\(x\) increased by twice the square of \(x\).

Write \(y\) as a function of \(x,\) where the value of \(y\) is equal to the given expression.The reciprocal of the cube of \(x\).

Which of the following relations are also functions? Explain. $$y^{2}=3 x-5$$

Write \(y\) as a function of \(x,\) where the value of \(y\) is equal to the given expression.Two-thirds of the amount by which \(x\) exceeds 4.

Which of the following relations are also functions? Explain. $$2 x^{2}=3 y^{2}-4$$

Write the equation called for in each of the following statements. Refer to Appendix \(A,\) "Summary of Facts and Formulas," if necessary.Express the area \(A\) of a triangle as a function of its base \(b\) and altitude \(h\).

Write the equation called for in each of the following statements. Refer to Appendix \(A,\) "Summary of Facts and Formulas," if necessary.Express the hypotenuse \(c\) of a right triangle as a function of its legs, \(a\) and \(b\).

Which of the following relations are also functions? Explain. Is the set of ordered pairs (1,3),(2,5),(3,8),(4,12) a function? Explain.

8\. Express the volume \(V\) of a sphere as a function of the radius \(r\) 9\. Express the power \(P\) dissipated in a resistor as a function of its resistance \(R\) and the current \(I\) through the resistor.

Which equations are in explicit form and which in implicit form? $$y=5 x-8$$

Write the equation called for in each of the following statements. Refer to Appendix \(A,\) "Summary of Facts and Formulas," if necessary.A car is traveling at a speed of 55 mi/h. Write the distance \(d\) traveled by the car as a function of time \(t\).

Substitute the literal values into each function. $$\text { If } f(x)=2 x^{2}+4, \text { find } f(a)$$.

Label the variables in each equation as dependent or independent. $$y=3 x^{2}+2 x$$

Label the variables in each equation as dependent or independent. $$x=3 y-8$$

Substitute the literal values into each function. $$\text { If } f(x)=5 x+1, \text { find } f(a+b)$$.

Label the variables in each equation as dependent or independent. $$w=3 x+2 y$$

Substitute the literal values into each function. $$\text { If } f(x)=5-13 x, \text { find } f(-2 c)$$.

Substitute the sets of values into each function. $$\text { If } f(x, y)=3 x+2 y^{2}-4, \text { find } f(2,3)$$.

$$\text { If } f(x, y)=y-3 x, \text { find } 3 f(2,1)+2 f(3,2)$$

Rewrite the following implicit equations in the explicit form, \(y=f(x)\). $$x+y=5$$

Substitute the sets of values into each function. $$\text { If } g(a, b)=2 b-3 a^{2}, \text { find } g(4,-2)$$.

Rewrite the following implicit equations in the explicit form, \(y=f(x)\). $$2 x-y+4=0$$

Substitute the sets of values into each function. $$\text { If } f(x, y)=y-3 x, \text { find } 3 f(2,1)+2 f(3,2)$$.

Manipulating Functions.If \(y=5 x+3,\) write \(x=f(y)\).

Substitute the given numerical value into each function. $$\text { If } f(x)=2 x^{2}+4, \text { find } f(3)$$

Manipulating Functions.$$\text { If } y=\frac{1}{x}-\frac{1}{5}, \text { write } x=f(y)$$.

Substitute the given numerical value into each function. $$\text { If } f(x)=5 x+1, \text { find } f(1)$$

Manipulating Functions.$$\text { If } x^{2}+y=x-2 y+3 x^{2}, \text { write } y=f(x)$$.

Substitute the given numerical value into each function. $$\text { If } f(x)=15 x+9, \text { find } f(3)$$

Manipulating Functions.$$\text { If } 5 p-q=q-p^{2}, \text { write } q=f(p)$$.

Substitute the given numerical value into each function. $$\text { If } g(x)=9-3 x^{2}, \text { find } g(-2)$$

Manipulating Functions.The power \(P\) dissipated in a resistor is given by \(P=I^{2} R .\) Write \(R=f(P, I)\).

Substitute the given numerical value into each function. $$\text { If } g(x)=9-3 x^{2}, \text { find } g(-2)$$

Manipulating Functions.Young's modulus \(E\) is given by $$E=\frac{P L}{a e}$$.Write \(e=f(P, L, a, E)\).

Substitute the given numerical value into each function. $$\text { If } h(x)=x^{3}-2 x+1, \text { find } h(2.55)$$

Composite Functions.Given the functions \(g(x)=2 x+3\) and \(f(x)=x^{2},\) write the composite function \(g[f(x)]\).

Substitute the given numerical value into each function. $$\text { If } f(x)=7+2 x, \text { find } f(3)$$

Composite Functions.Given the functions \(g(x)=x^{2}-1\) and \(f(x)=3+x,\) write the composite function \(f[g(x)]\).

Substitute the given numerical value into each function. $$\text { If } f(x)=x^{2}-9, \text { find } f(-2)$$

Composite Functions.Given the functions \(g(x)=1-3 x\) and \(f(x)=2 x,\) write the composite function \(g[f(x)]\).

Substitute the given numerical value into each function. $$\text { If } f(x)=2 x+7, \text { find } f(-1)$$

Composite Functions.Given the functions \(g(x)=x-4\) and \(f(x)=x^{2},\) write the composite function \(f[g(x)]\).

Applications. The distance traveled by a freely falling body is a function of the elapsed time \(t:\) $$ f(t)=v_{0} t+\frac{1}{2} g t^{2} \quad \text { ft } $$ where \(v_{0}\) is the initial velocity and \(g\) is the acceleration due to gravity \(\left(32.2 \mathrm{ft} / \mathrm{s}^{2}\right)\) If \(v_{0}\) is \(55.0 \mathrm{ft} / \mathrm{s},\) find \(f(10.0), f(15.0),\) and \(f(20.0)\)

Given \(g(x)=x^{3}\) and \(f(x)=4-3 x,\) find $$f[g(x)]$$

Applications. The resistance \(R\) of a conductor is a function of temperature: $$ f(t)=R_{0}(1+\alpha t) $$ where \(R_{0}\) is the resistance at \(0^{\circ} \mathrm{C}\) and \(\alpha\) is the temperature coefficient of resistance \(\left(0.00427 \text { for copper). If the resistance of a copper coil is } 9800 \Omega \text { at } 0^{\circ} \mathrm{C}\right.\) find \(f(20.0), f(25.0),\) and \(f(30.0)\)

Given \(g(x)=x^{3}\) and \(f(x)=4-3 x,\) find $$f[g(3)]$$